GEOMETRIA FRACTAL, OSCILAÇÃO E SUAVIDADE DE FUNÇÕES CONTÍNUAS

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1 Universidade de Aveiro Departamento de Matemática 2008 JOSÉ ABEL LIMA CARVALHO GEOMETRIA FRACTAL, OSCILAÇÃO E SUAVIDADE DE FUNÇÕES CONTÍNUAS FRACTAL GEOMETRY, OSCILLATION AND SMOOTHNESS OF CONTINUOUS FUNTIONS

2 Universidade de Aveiro Departamento de Matemática 2008 JOSÉ ABEL LIMA CARVALHO GEOMETRIA FRACTAL, OSCILAÇÃO E SUAVIDADE DE FUNÇÕES CONTÍNUAS FRACTAL GEOMETRY, OSCILLATION AND SMOOTHNESS OF CONTINUOUS FUNTIONS Tese apresentada à Universidade de Aveiro para cumprimento dos requisitos necessários à obtenção do grau de Doutor em Matemática, realizada sob a orientação científica do Dr. António Manuel Rosa Pereira Caetano, Professor Associado com Agregação do Departamento de Matemática da Universidade de Aveiro Apoio financeiro do POCTI no âmbito do III Quadro Comunitário de Apoio. Co-financiamento do FEDER. Apoio financeiro do convénio GRICES- DAAD. Co-financiamento do grupo Fractal analysis da Universidade de Jena. Apoio financeiro da FCT e do FSE no âmbito do III Quadro Comunitário de Apoio. Bolsa de Investigação SFRH/BD/16029/2004. Apoio financeiro da Unidade de Investigação Matemática e Aplicações da Universidade de Aveiro.

3 Dedicado à mári

4 o júri presidente Prof.ª Dr.ª Maria Helena Nazaré reitora da Universidade de Aveiro Prof. Dr. Stéphane Jaffard professeur de Mathématiques, Laboratoire d Analyse et de Mathématiques Appliquées, Université Paris XII, França Prof. Dr. António Manuel Rosa Pereira Caetano professor associado com agregação da Universidade de Aveiro Prof.ª Dr.ª Susana Margarida Pereira da Silva Domingues de Moura professora auxiliar da Faculdade de Ciências e Tecnologia da Universidade de Coimbra Prof. Dr. José Alexandre da Rocha Almeida professor auxiliar da Universidade de Aveiro

5 agradecimentos Gostaria de agradecer ao Professor Caetano especialmente pela escolha do tema. Também por ter lido cuidadosamente todas as versões preliminares, pelas centenas de correcções do inglês e por muitos comentários preciosos para a melhoria e o enriquecimento do texto. E pelo excelente relacionamento de trabalho. Gostaria também de agradecer ao Professor Triebel pelos seus valiosos conselhos e sugestões no que concerne a notações e estrutura geral. E também pela sua enorme disponibilidade quando seja necessário discutir ou clarificar ideias. Um agradecimento especial ao Paolo Vettori pela grande ajuda no uso do tex. acknowledgements I would like to thank Professor Caetano, who read carefully all preliminary versions of the work, for the choice of the theme, motivation, hundreds of corrections in the English, many valuable comments for improvement and enrichment of the text, as well as, the excellent working environment. I would like also to thank Professor Triebel for his valuable advices and suggestions concerning the notations and the layout of the work, as well as his precious availability during the discussions and clarification of ideas. I am also indebted to Paolo Vettori for the great help concerning the tex.

6 palavras-chave espaços de Besov, suavidade, levantamento, espaços de oscilação, dimensões de caixa e de Hausdorff, funções chirp e de Weierstrass, medidas e conjuntos-h, espaços em conjuntos-d. resumo Primeiramente, relacionamos as dimensões superior e inferior de caixa com os espaços de oscilação e desenvolvemos imersões entre os espaços de oscilação e os espaços de Besov. Então, obtemos valores maximais e minimais para as dimensões de caixa e de Hausdorff, sobre todas as funções contínuas e compactamente suportadas em R n, com integrabilidade 0 < p e suavidade exacta s > 0. Calculamos também as dimensões de caixa e de Hausdorff para certas funções chirp e do tipo Weierstrass e assim, usando o operador levantamento para testar o comportamento das dimensões em função da suavidade, mostramos que há uma certa incerteza na relação entre suavidade e dimensões de gráficos. Seguidamente, investigamos alguns critérios para decidir quando é que um gráfico de uma função não é um conjunto-d. Além disso, para cada d entre n e n +1, construímos uma função sobre [0,1] n cujo gráfico é um conjunto-d. E na classe das funções reais definidas sobre um conjunto-h 1 contido na recta real, provamos a existência de funções cujos gráficos são conjuntos-h, sob condições apropriadas para a função de massa h. Em particular, obtemos os valores maximais para as dimensões de caixa e de Hausdorff para todos os gráficos de funções de Hölder sobre um conjunto-d 1 contido na recta real, assim como o valor maximal para a dimensão de Hausdorff para todas as funções de B s p,q que sejam traços de funções contínuas num tal conjunto-d 1, para qualquer 1< p < e 0< s < d 1. Finalmente, redireccionamos o nosso interesse para as aplicações práticas baseadas nas imersões entre os espaços de oscilação e os de Besov, e consideramos o problema geral da detecção de sinal, apresentando uma simulação numérica para sinais do tipo onda e para sinais chirp.

7 keywords Besov spaces, smoothness, lifting, oscillation spaces, box and Hausdorff dimensions, Weierstrass and chirp functions, measures and h-sets, spaces on d-sets. abstract Firstly, we relate upper and lower box dimensions with oscillation spaces and we develop embeddings or inclusions between oscillation spaces and Besov spaces. Then, we obtain maximal and minimal box and Hausdorff dimensions over all continuous real functions compactly supported on R n, with integrability 0 < p and exact smoothness s > 0. We also calculate box and Hausdorff dimensions of some chirp and Weierstrass-type functions and then, by using the lifting operator to check the behavior of dimensions against smoothness, we show that there is some uncertainty in the relation between smoothness and dimensions of the graphs. Secondly, we investigate some criteria in order to decide when the graph of a function is not a d-set. Moreover, for each d between n and n +1, we construct a function on [0,1] n the graph of which is a d-set. And in the class of the real functions defined on a h 1 -set contained in the real line, we prove the existence of functions whose graphs are h-sets, under appropriate assumptions for the mass function h. In particular, we obtain maximal box and Hausdorff dimensions of the graphs of Hölder functions on a d 1 -set contained in the real line, and the maximal Hausdorff dimension over all functions of B s p,q which are traces of continuous functions on a d 1 -set, for any 1< p < and 0< s < d 1. Finally, we switch to practical applications based on the embeddings between oscillation and Besov spaces, and consider the problem of the general detection, presenting a numerical simulation for wave and chirp signals.

8 palavras-chave espaços de Besov, suavidade, levantamento, espaços de oscilação, dimensões de caixa e de Hausdorff, funções chirp e de Weierstrass, medidas e conjuntos-h, espaços em conjuntos-d. resumo Primeiramente, relacionamos as dimensões superior e inferior de caixa com os espaços de oscilação e desenvolvemos imersões entre os espaços de oscilação e os espaços de Besov. Então, obtemos valores maximais e minimais para as dimensões de caixa e de Hausdorff, sobre todas as funções contínuas e compactamente suportadas em R n, com integrabilidade 0 < p e suavidade exacta s > 0. Calculamos também as dimensões de caixa e de Hausdorff para certas funções chirp e do tipo Weierstrass e assim, usando o operador levantamento para testar o comportamento das dimensões em função da suavidade, mostramos que há uma certa incerteza na relação entre suavidade e dimensões de gráficos. Seguidamente, investigamos alguns critérios para decidir quando é que um gráfico de uma função não é um conjunto-d. Além disso, para cada d entre n e n +1, construímos uma função sobre [0,1] n cujo gráfico é um conjunto-d. E na classe das funções reais definidas sobre um conjunto-h 1 contido na recta real, provamos a existência de funções cujos gráficos são conjuntos-h, sob condições apropriadas para a função de massa h. Em particular, obtemos os valores maximais para as dimensões de caixa e de Hausdorff para todos os gráficos de funções de Hölder sobre um conjunto-d 1 contido na recta real, assim como o valor maximal para a dimensão de Hausdorff para todas as funções de B s p,q que sejam traços de funções contínuas num tal conjunto-d 1, para qualquer 1< p < e 0< s < d 1. Finalmente, redireccionamos o nosso interesse para as aplicações práticas baseadas nas imersões entre os espaços de oscilação e os de Besov, e consideramos o problema geral da detecção de sinal, apresentando uma simulação numérica para sinais do tipo onda e para sinais chirp.

9 keywords Besov spaces, smoothness, lifting, oscillation spaces, box and Hausdorff dimensions, Weierstrass and chirp functions, measures and h-sets, spaces on d-sets. abstract Firstly, we relate upper and lower box dimensions with oscillation spaces and we develop embeddings or inclusions between oscillation spaces and Besov spaces. Then, we obtain maximal and minimal box and Hausdorff dimensions over all continuous real functions compactly supported on R n, with integrability 0 < p and exact smoothness s > 0. We also calculate box and Hausdorff dimensions of some chirp and Weierstrass-type functions and then, by using the lifting operator to check the behavior of dimensions against smoothness, we show that there is some uncertainty in the relation between smoothness and dimensions of the graphs. Secondly, we investigate some criteria in order to decide when the graph of a function is not a d-set. Moreover, for each d between n and n +1, we construct a function on [0,1] n the graph of which is a d-set. And in the class of the real functions defined on a h 1 -set contained in the real line, we prove the existence of functions whose graphs are h-sets, under appropriate assumptions for the mass function h. In particular, we obtain maximal box and Hausdorff dimensions of the graphs of Hölder functions on a d 1 -set contained in the real line, and the maximal Hausdorff dimension over all functions of B s p,q which are traces of continuous functions on a d 1 -set, for any 1< p < and 0< s < d 1. Finally, we switch to practical applications based on the embeddings between oscillation and Besov spaces, and consider the problem of the general detection, presenting a numerical simulation for wave and chirp signals.

10 Contents 1 Introduction Motivation Main aims Definitions Besov spaces and class s Oscillation spaces Lower counterpart classes Upper and lower box dimensions Measures, h-sets, and Hausdorff dimension Maximal and minimal dimensions Dimensions and smoothness in function spaces Summary of the Chapter Box dimensions in oscillation spaces Embeddings between Besov and oscillation spaces Functions for extremal box dimensions Summarizing maximal and minimal dimensions Lifting operator and dimensions Fractal geometry of Weierstrass-type functions Summary of the Chapter Dimensions and smoothness of some graphs Comparing box and Hausdorff dimensions Search and construction of d-sets Existence and construction of h-sets Longer proofs and explanations Proofs of Section 3.2 and Theorem Proofs of Section Proofs of Section

11 5.4 Proofs of Section 3.5 (except Theorem 3.5.3) Proofs of Section 4.2 and Theorem Proofs of Section Proofs of Section Discussion, applications and open problems Summary of the Chapter Some complementary results Comparison of different techniques Development of applications Open problems and conjectures

12 List of Figures Figure 1: Typical curve s f (t) for t 0, given a compactly supported distribution f S (R n ) \ C (R n ) 6 Figure 2: Graph of a real function f with a covering by small dyadic squares 7 Figure 3: The sequence (F ϕ j ) j N0 is an infinitely smooth resolution of the unity in frequency 14 Figure 4: Representation of dyadic cubes (squares) in the plane R 2 17 Figure 5: The oscillation of f over I = [0, 0.5] is osc I (f) = sup I f inf I f 17 Figure 6: The Corollary gives Dim s p n + 1 min{1, s} for 0 < p and s > n p 30 Figure 7: The Corollary gives dim s p n + 1 s max{1, p} for 0 < p < and 0 < s < 1 max{1,p} 31 Figure 8: The Theorem gives Dim s p = n + 1 for 0 < p < and 0 < s n p 33 Figure 9: The Theorem gives the exact values of Dim s p for 0 < p and s > 0 39 Figure 10: The Theorem gives the exact values of dim s p for 0 < p and s > 0 39 Figure 11: Case n p < 1. The solid line represents Dims p and the dashed line represents dim s p. For the case n p > 1 the solid line is simpler 40 Figure 12: Graph of the Weierstrass function W s (x) = j 1 2 js sin(2 j x) with s = Figure 13: Graph of f s (x) = j 1 2 ν js ζ 0 (2 ν j x), where 0 < s < 1, ν j = 2 jκ with κ N such that 2 κ s > 1, and ζ 0 C 1 (R) is defined by ζ 0 (x) = 1 k Z ω(x k), with ω(x) = e 1 4x 2 if x < 1 2 and ω(x) = 0 otherwise 58 Figure 14: Graph of W s (x) = j 1 ϱ js Λ(ϱ j x), with ϱ = 2k + 1 according to Corollary with Definition We show the case ϱ = 9 and s = 0.6, with a triangular wave Λ62 Figure 15a: According to Definition we start with W (0) V,H. We choose V = 4 64 Figure 15b: First iteration W (1) V,H, according to Definition Here V = 4, H 0 = 3 65 Figure 15c: Second iteration W (2) V,H, according to Definition 4.4.2, with V = 4, H 0 =

13 Figure 16: Asymptotical qualitative growth of µ 0 (B r ) on Γ(f s K ), where 0 < B r = r < 1 and 0 < s < 1. Points of minimal mass occur for r = r j j 2 ν j, with µ 0 (B j ) j rj 2 s, and points of maximal mass occur for r = r j j 2 ν jσ, with µ 0 (B j ) j r 2 Ξ(2κ,s) j = r 2 s σ j 68 Figure 17: Graph of the restriction W s K to a Cantor-like set, according to the Proof of Theorem 4.2.7, where W s (x) = j 1 ρ γjs g(ρ γj x + θ j ) 114 Figure 18a: This graph shows the general aspect of a h-set, W := W h, when constructed as in Definition recall Figures 15abc and generalized according to the Proof of Theorem Figure 18b: The graph W K := T K W originated from the one of Figure 18a, by applying the domain operator T K, according to Definition (We don t see the details at a very small scale.) 129 Figure 19: Signal with added standard gaussian noise; sample dimension k 0 = Figure 20a: Wave-signal detection - Fourier method (Runovski, [27]) 149 Figure 20b: Wave-signal detection - Oscillation method 149 Figure 21a: Chirp-signal detection - Fourier method (Runovski, [27]) 150 Figure 21b: Chirp-signal detection - Oscillation method 150 Tables summarizing maximal and minimal dimensions 42 4

14 1 Introduction 1 Introduction 1.1 Motivation The filling of the gaps between Sobolev spaces, giving origin to the concept of generalized smoothness s, for any real s instead of integer only, was an important step in the long way of systematization and unification of the function spaces. As a major contribution in this line of research, since the Sixties of the last century have been written many papers about Besov spaces Bp,q(R s n ) and Triebel-Lizorkin spaces Fp,q(R s n ), so the function spaces theory started to be one self-contained branch of the functional analysis. The diversification of techniques to measure smoothness, a fundamental and rich topic closely related with the development of function representation techniques, was motivated by theoretical and practical reasons and has been essential in that view of unification. From the beginning, the concept of smoothness in the theory of function spaces has been closely connected to the graphical interpretation, one of the main sources of inspiration, enthusiasm and motivation for mathematicians, and more recently, many strong and popular techniques in theory of function spaces still intimately related to methods in fractal geometry. Consider n as a natural number, A as a non-empty subset of R n, and let f : A R be a real valued function. As usually, the graph of f, denoted by Γ(f), is the subset of R n R R n+1 containing all pairs (x, f(x)) when x runs over all x A. Having in mind the preparation and development of tools which we want able to capture the intrinsic and unique properties of the graphs, we will relate three different forms of measuring smoothness of continuous real functions: The frequency structure, in the sense of the Besov spaces that contain f, or not; Box dimension of Γ(f), that depends on the number of cubes needed to cover the graph Γ(f) (see Figure 2 for the case n = 1); The oscillations between local maximums and minimums of the image f(x). The frequency structure of f is characterized by a suitable resolution of the unity in 5

15 1.1 Motivation 1 Introduction frequency and weighted summation of the resolution parts. This is the idea in the basis of the definition of the Besov spaces B s p,q(r n ) and the Triebel-Lizorkin spaces F s p,q(r n ). Let 0 < p (p < in the F -case), 0 < q and s R. We would like to recall (see [31], 2.3.5) that the spaces B s p,q(r n ) and F s p,q(r n ) contain as special cases many well-known spaces, as the Bessel potential spaces (fractional Sobolev spaces), Hölder-Zygmund spaces, and Hölder spaces, H s p(r n ) = F s p,2(r n ) if 1 < p < and s R, C s (R n ) = B s, (R n ) if s > 0, C s (R n ) = B s, (R n ) if 0 < s integer. (Here, we use the notation = to mean equivalent norms on the spaces considered.) Consider now 0 < p and s R. We define the class B s p (R n ) of those elements f S (R n ) with an integrability p and an exact smoothness s. Recall that S (R n ) is the class of all tempered distributions, the topological dual of the Schwartz class S(R n ) of all infinitely smooth and rapidly decreasing complex valued functions. More precisely, f is an element of the class B s p (R n ) if and only if it satisfies the relations f B σ p,q(r n ) if σ < s and f B σ p,q(r n ) if σ > s. The q is omitted in the notation, since this definition does not depend on q. Moreover, equivalently we can write that f B s p (R n ) sup{σ R : f B σ p,q(r n )} = s. We will use an useful graphical representation in the ( 1, s)-plane, interpreted in the s p sense, i.e. each point ( 1, s) symbolizes any tempered distribution belonging to Bs p p (R n ). Furthermore, given a compactly supported f S (R n ) \ C (R n ), we define s f (t) as the real number such that f B s f (t) p (R n ), where t := 1 p 0. Therefore, the graph of s f (t) describes (see [35], p. 414) a continuous, concave and non-decreasing curve with left hand derivative s f (t) n for t > 0. s Figure 1: Typical curve s f (t) for 1/p t 0, given a compactly supported distribution f S (R n ) \ C (R n ) 6

16 1.1 Motivation 1 Introduction We are specially interested in continuous real functions with a compact support on R n. Actually, the class B s p (R n ) does not contain functions satisfying these conditions if s < 0, since it follows by elementary calculations that if f is such a function then f B ε p,q(r n ), ε > 0. But, as soon as we restrict ourselves to s > 0, then by the following Theorem that class contains a non-empty subset of continuous real functions with a compact support. Theorem ([37], p. 202, Corollary 9, with elementary adaptations) Consider s > 0 and let f s be as in [37], p Then f s B s p (R n ) for 0 < p. Though the existence result of the Theorem can be extended to the case when s equals zero, in the rest of the work we will consider only s > 0, since we do not loose any generality by ruling out that limiting case. The class B s p (R n ) will play an important role in all the work, specially as far as concerns continuous real functions with a compact support. With the corresponding notation applied to the spaces we mentioned above, and giving us a good reason to restrict the attention to the Besov scale, we have the identities Fp s (R n ) = Bp s (R n ) if 0 < p < and s R, H s p (R n ) = B s p (R n ) if 1 < p < and s R, and C s (R n ) = C s (R n ) = B s (R n ) if s > 0. f(x) x Figure 2: Graph of a real function f with a covering by small dyadic squares 7

17 1.1 Motivation 1 Introduction Let us introduce now some concepts about fractal geometry, which we want to relate with the mentioned function spaces. By definition, a dyadic cube I in R n with volume I = 2 νn is a set of the form I := 2 ν ([0, 1] n + k), where k Z n and ν N 0. Consider f : A R n R as a continuous function such that Γ(f) is a non-empty and bounded subset of R n+1, and let M(ν, f) be the minimum number of dyadic cubes in R n+1, with volume 2 ν(n+1), required to cover Γ(f) (see Figure 2 for the case n = 1). We define the upper box dimension of Γ(f), log dim B Γ(f) := lim 2 M(ν, f) ν. ν The lower box dimension of Γ(f), dim B Γ(f), is defined in a similar way but with the lower limit. For a continuous function f : R n R we take dim B Γ(f) := sup k N dim B Γ(f [ k,k] n) and similarly for the lower case. If the upper box dimension and lower box dimension coincide then we define the box dimension of Γ(f), dim B Γ(f), as the common value of them. The oscillations of the graph of a function play an important role in the estimation of the box dimension. oscillation spaces. We characterize the oscillations of a real function f by the If I is a dyadic cube we define the oscillation of f over I by osc I (f) := sup I f inf I f. Given a real number α R, the oscillation space V α (R n ) contains all real measurable functions with f V α (R n ) := sup 2 ν(α n) osc I (f) <. ν 0 I =2 νn The parameter α is related with the upper box dimension dim B Γ(f). To see this in an heuristic way, suppose that f V α (R n ) for a given α > 0. The larger the α > 0, the weaker the faster oscillations of f, and this means more smoothness and easiness in covering Γ(f) with small cubes (or balls), i.e. Γ(f) has a smaller upper box dimension. Hence, the parameter α is a measure of smoothness in the oscillation spaces. In the same way, the parameter s measures smoothness in Besov spaces. In fact, the frequency structure has a direct influence in the oscillations of a function, therefore Besov spaces B s p,q(r n ) and oscillation spaces V α (R n ) are related. Assume that f B s p,q(r n ). The larger the s > 0, the weaker the contribution of the higher frequencies in the 8

18 1.2 Main aims 1 Introduction composition of f. As a consequence, the fast oscillations have a smaller amplitude and this indicates that f V α (R n ) for a large α > 0. A converse qualitative implication is also valid. This will be made more precise in Theorems and below. Besides the box dimension, there are many other forms of dimension (see [14], Chapters 2 and 3), in particular the Hausdorff dimension (see [14], pp ), denoted by dim H, which is the oldest and one of the most distinguished. A major advantage of this dimension is that it derives from the Hausdorff s measures and so it satisfies the countable stability property, which fails for the box dimension. If F 1, F 2,... is a sequence of sets, dim H i 1 F i = sup i 1 dim H F i, but the counterpart for box dimension does not hold. An example that shows how distinct box and Hausdorff dimensions can be, is the graph of the chirp function ch r (x) := x r cos(e 1/x ), 0 < x < 1, where r 0. It holds dim H Γ(ch r ) = 1, because ch r has a bounded derivative on [1/k, 1) for any k N. However, dim B Γ(ch r ) = 2. (In Definitions 3.4.1/3.4.2 below, we will constructed other extremal bizarre functions based precisely in this kind of chirp functions.) Box dimension is one of the most widely used dimensions in practice, perhaps due to its relative easiness of mathematical calculation or computational estimation. And as we will see in Section 4.4 below, in some cases we can directly take advantage of the box dimension in order to get estimations for the much more difficult Hausdorff dimension, due to the inequalities n dim H Γ(f) dim B Γ(f) dim B Γ(f) (see e.g. [14], p. 43). 1.2 Main aims One of the main aims of this thesis is to calculate the maximal and minimal dimensions, Dim s p and dim s p, for 0 < p and s > 0. We define the maximal dimension by Dim s p := sup dim B Γ(f), f Bp s (Rn ) where the supremum is taken over all continuous real functions f B s p (R n ). For the minimal dimension, dim s p := inf f B s p (R n ) dim B Γ(f), 9 we take the infimum over all real

19 1.2 Main aims 1 Introduction functions f B s p (R n ) continuous and with a compact support. Later on, see remark (b) of Definition 2.6.1, we will justify the addition of this compact support condition. The following Theorem provides a lower bound for the minimal dimension if n = 1, p = 1 and s > 0. In [13], p. 220, we find an analogous result but, as we will see in the comments that follow Theorem 3.4.1, both Theorem and the counterpart in [13], p. 220, cannot be improved with the other inequality. Theorem Consider s > 0 and let f : R R be a continuous function with a compact support. Assume in addition that f belongs to the class B s 1 (R). Then we have dim B Γ(f) 2 min{1, s}. In order to prove our main results, we need to develop relations between the upper box dimension, the oscillation spaces V α (R n ) and the Besov spaces B s p,q(r n ), namely we need embeddings or inclusions between these spaces. Similarly, we develop some corresponding results for the lower box dimension. Then, we intend to make a characterization of the Besov-Triebel-Lizorkin spaces namely in terms of box dimensions, which is developed in Section 3 below. In a quite natural way, several questions arise from such a fractal-oriented research, whose process of answering leads to the main aims and results of this work, particularly the followings: Characterization of the oscillation spaces in terms of upper box dimension; To establish embeddings between Besov spaces and oscillation spaces; To develop lower counterpart results in order to deal with the lower box dimension; Estimation of maximal and minimal dimensions over Besov spaces or classes, firstly for continuous real functions on R n, and afterwards by considering traces of such functions on a d 1 -set (though under some restrictions at first); To show that there exists uncertainty attached to the behavior of dimensions against smoothness, by applying the lifting operator; To calculate the exact smoothness and box and Hausdorff dimensions of the graphs of particular classes of functions, namely of some Weierstrass-type functions; To make distinction between smoothness and dimensions, as well as between the 10

20 2 Definitions various types of dimensions; Development of criteria in order to decide when the graphs of special Weierstrasstype functions are not d-sets or h-sets. Construction of graphs which are d-sets and even h-sets, both on R n and on a h 1 -set (under appropriate sufficient conditions); To compare different approach techniques, namely wavelet, frequency, and nonsmooth atoms approaches; Development of academic and practical applications. 2 Definitions We start by giving some basic definitions and by recalling some classic spaces of functions, as well as some well-known operators on tempered distributions. After this preliminary and necessary background, we introduce the Besov-Triebel-Lizorkin spaces and the class B s p (R n ) of all tempered distributions with integrability p and exact smoothness s. The several appropriate but non-important strictly positive constants are represented by c. When in the same expression they are distinguished by c, c, c,... Frequently these constants have an index, for example c R depends on R. is a constant that 2.1 Besov spaces and class s Definition (a) Let x := (x 1,..., x n ) R n. Then x := n i=1 x2 i is the Euclidean norm of x, and B ρ (x) := {t R n : t x ρ} the closed ball centered at x and with radius ρ > 0. Furthermore, [ ] stands for the integer part of a positive number. We consider also here the scalar product ξx := n i=1 ξ ix i, where ξ := (ξ 1,..., ξ n ) R n. Moreover, we will use also the notation for the absolute value of a complex number. Remark: We will use frequently in the text, see for instance Definitions 2.5.5/2.5.6 below, a similar but different notation for balls and respective radii or diameters. (b) Let α := (α 1,..., α n ) (N 0 ) n be a multi-index. Then D α is the classic derivative 11

21 2.1 Besov spaces and class s 2 Definitions operator. We define here the class C (R n ) := {ϕ : R n C: D α ϕ is bounded and continuous for all multi-indexes α (N 0 ) n }, as well as the class C (R n ): ϕ has a compact support}. C 0 (R n ) := {ϕ Definition The Schwartz class, S(R n ), is the family of functions defined by S(R n ) := {ψ : R n C with ψ C (R n ) and p r,s (ψ) <, r, s N 0 }, where p r,s (ψ) := sup x R n sup α s x r D α ψ(x), the inner supremum taken over all multi-indexes α (N 0 ) n with α s. The topology of S(R n ) is defined by this family of semi-norms p r,s with r, s N 0. Definition We define, as usually, the class S (R n ) of all tempered distributions as the dual of S(R n ), equipped with the weak topology or with the strong topology. Definition Let e be the Neper s number and consider i = 1. We consider (see e.g. [31], p. 13) F : S(R n ) S(R n ), the Fourier transform on S(R n ), defined by (F ψ)(ξ) := 1 (2π) n 2 R n ψ(x)e iξx dx, ξ R n, and in the standard way we extend consistently the definition of Fourier transform to F : S (R n ) S (R n ) by < F T, ψ > := < T, F ψ >, ψ S(R n ). Remark: By [22], p. 25, with some elementary adaptations, we have on S(R n ) the identity (F 1 ψ)(ξ) = 1 (2π) n 2 R n ψ(x)e iξx dx, ξ R n, and on S (R n ) we have the property < F 1 T, ψ > = < T, F 1 ψ >, ψ S(R n ). Definition ([31], p. 58) Given σ R, we define the lifting operator I σ : S (R n ) S (R n ), as usually, by I σ f := F 1 (1 + 2 ) σ 2 F f. (For sake of simplicity we write F 1 gf f with the meaning F 1 (g (F f)), whenever f and g are elements of S (R n ). Furthermore, the reader should be aware that we will deal only with the case where f is a continuous real function.) Remark: As it is well known that F and F 1 are continuous on S (R n ), with both the strong and the weak topologies, then the lifting operator I σ is also continuous on S (R n ) for those 12

22 2.1 Besov spaces and class s 2 Definitions topologies. Definition Let 0 < p. Thus, we define the Lebesgue spaces where L p (R n ) := {f : R n C measurable with f Lp(R n ) < }, f Lp (R n ) := ( R n f(t) p dt ) 1/p. ( f L (R n ) := ess-sup f.) Definition Consider a bounded set G R n. It will be convenient to consider also S G (R n ) := {ψ S(R n ) with supp(f ψ) G}, as well as L G p (R n ) := {f L p (R n ) with supp(f f) G} for 0 < p. The preliminary definitions above allow us to introduce the Besov spaces and the Triebel-Lizorkin spaces by considering the frequency approach. Later on we will make use also of the wavelet approach, for instance in order to make shorter and easier some proofs as well as for a matter of comparison of approaches. Definition Let F be the Fourier transform on S(R n ), and let be the convolution operator on S (R n ). Let ϕ S(R n ) with supp(f ϕ) {ξ R n : 1/2 ξ 2}, and (F ϕ)(ξ) + (F ϕ)(2 1 ξ) = 1 if 1 ξ 2. For all j N we define ϕ j (x) := 2 jn ϕ(2 j x), x R n. Let ϕ 0 S(R n ) with supp(f ϕ 0 ) {ξ R n : ξ 2}, and (F ϕ 0 )(ξ)+(f ϕ)(2 1 ξ) = 1 if ξ 2. Consider 0 < p, 0 < q and s R. We define the classic Besov spaces B s p,q(r n ) := {f S (R n ) with f B s p,q (R n ) < }, where f B s p,q (R n ) := ( j 0 ) 1/q ( ) 2 js q ϕ j f Lp (R n ). (Modification f B s p, (R n ) := sup j 0 2 js ϕ j f Lp(R n ).) 13

23 2.1 Besov spaces and class s 2 Definitions Consider 0 < p <, 0 < q and s R. We define also the classic Triebel- Lizorkin spaces F s p,q(r n ) := {f S (R n ) with f F s p,q (R n ) < }, where f F s p,q (R n ) := ( j 0 (2js ϕ j f ) q ) 1/q Lp (R n ). (Usual modification for q =.) 1 Fϕ 0 (ξ) Fϕ 1 (ξ) Fϕ 2 (ξ) Fϕ 3 (ξ) ξ Figure 3: The sequence (F ϕ j ) j N0 is an infinitely smooth resolution of the unity in frequency It seems convenient to define also the Besov class with integrability p and smoothness s- Bp s (R n ) := {f S (R n ) such that f Bp,q s ε (R n ), ε > 0}, the Besov class with integrability p and smoothness s+ Bp s+ (R n ) := {f S (R n ) such that f Bp,q s+ε (R n ) for some ε > 0}, and the Besov class with integrability p and exact smoothness s or, shortly, class s as Remark: B s p (R n ) := B s p (R n ) \ B s+ p (R n ). (a) We notice that the functions ϕ and ϕ 0 exist. This is clear if we define first (F ϕ)(ξ) for ξ [1/2, 1], with (F ϕ)(ξ) := 0 (respectively (F ϕ)(ξ) := 1) for ξ in a neighborhood of 1/2 (respectively 1) see comments before Lemma After we define (F ϕ)(ξ) := 1 (F ϕ)(2 1 ξ) for ξ [1, 2], and (F ϕ 0 )(ξ) := 1 (F ϕ)(2 1 ξ) for ξ 2. (b) The sequence (F ϕ j ) j N is a resolution of the unity (in the frequency side). This means that we have the (pointwise) equality j 0 (F ϕ j)(ξ) = (F ϕ 0 )(ξ)+ j 1 (F ϕ)(2 j ξ) = 14

24 2.1 Besov spaces and class s 2 Definitions 1, ξ R n (see Figure 3). (c) Because of the property ϕ j f = (2π) n 2 F 1 ((F ϕ j )(F f)) with f S (R n ), then the spaces B s p,q(r n ) and F s p,q(r n ) above are the same (equivalent quasi-norms) Besov- Triebel-Lizorkin spaces defined in [31], p. 45. In particular, by [31], p. 46, the spaces B s p,q(r n ) and F s p,q(r n ) are independent (equivalent quasi-norms) of ϕ and ϕ 0. (d) Let 0 < p <, 0 < q 0, q 1, q 2, s R and ε > 0. Then by [31], p. 47, we have the embeddings B s+ε p,q 0 (R n ) F s p,q 1 (R n ) B s ε p,q 2 (R n ). Hence we obtain, in obvious notation, the identity Fp s (R n ) = Bp s (R n ). (e) Of course for 0 < p and s R we have the relations (identities and inclusions) B s+ p (R n ) = σ>s Bσ p,q(r n ) Bp,q(R s n ) σ<s Bσ p,q(r n ) = B s (R n ) and the counterparts for the spaces F s p (R n ) (0 < p < ). (f) Let 0 < p and s R. A precise characterization of the class B s p (R n ) is given by the equivalence f B s p (R n ) sup{σ R : f B σ p,q(r n )} = s. Equivalently, the right hand side condition can be written as f B σ p,q(r n ) if σ < s and f B σ p,q(r n ) if σ > s. (g) The classes Bp s (R n ), Bp s (R n ) and B s+ (R n ), particularly the later one, have here p definitions that do not coincide exactly with the given ones in [10, 11]. In the present work, however, we consider these classes as subsets of S (R n ) only for technical convenience. Furthermore, the notation B s+ p (R n ) suggests a class contained in B s p,q(r n ) because s+ denotes a smoothness non-smaller than s, in the same way that the notation Bp s (R n ) suggests a class that contains Bp,q(R s n ); having this in mind, the definitions given above for B s+ p property, as pointed out by previous part (e). (R n ) and B s (R n ) fulfil this intuitive and convenient p p Part (d) of the remark above justifies the independence of the class B s p (R n ) on the parameter q, and gives the identity B s p (R n ) = F s p (R n ) for 0 < p < and s R. With corresponding notation applied to other spaces, we have also the identities H s p (R n ) = B s p (R n ) if 1 < p < and s R, C s (R n ) = B s (R n ) if s > 0, 15

25 2.2 Oscillation spaces 2 Definitions and C s (R n ) = B s (R n ) if s > 0. These identities justify why we will restrict us to the Besov scale. Concerning examples of functions belonging to these classes, when s > 0, consider φ S(R) a real function with φ(0) 0, and define f s (x) = x s φ(x) if x 0, vanishing otherwise. Then we have f s C s (R) = B s (R), i.e., for p = the function f s has an exact smoothness s. Moreover, for all 0 < p and s > 0 the class B s p (R n ) contains a non-empty subset of continuous, real and compactly supported functions see Section 1.1, specially Theorem and comments before it, with reference to [35, 37]. 2.2 Oscillation spaces We define now a very different kind of linear spaces and the corresponding classes of smoothness. These are the oscillation spaces, whose smoothness index have an intimate conection with box dimensions. Later we will relate them with the Besov spaces. Definition Let N N\{1} and ν N 0. We call I a N-adic cube in R n if I can be written as Remark: I := n k=1 [(j k 1)N ν, j k N ν ], where j k {1,..., N ν }, k = 1,..., n. (a) I is a cube with side length N ν, contained in [0, 1] n. (b) We have I = N νn, where I represents the volume of I. Concerning the notation, see also remark of Definition (c) The integer N is not important and in what follows the reader may take for simplicity N = 2. This follows by remark (b) of Definition see also remark (b) of Definition and remark (a) of Definition Definition Let ν N 0. As a generalization of Definition 2.2.1, let I := N ν ([0, 1] n + k), with k Z n, be a cube of volume I = N νn (see Figures 2 and 4 for the case n = 2). We apply also to I the designation of N-adic cube. When N = 2 we use also the designation of dyadic for 2-adic. 16

26 2.2 Oscillation spaces 2 Definitions x 2 x 1 Figure 4: Representation of dyadic cubes (squares) in the plane R 2 Definition Consider ν N 0 and let f : [0, 1] n R be a function. We define (a) The oscillation of f over I, osc I (f) := sup I f inf I f, for all I as in Definition (see Figure 5). (b) Osc(ν, f) := I =N osc νn I (f), where the sum is taken over all I as in Definition f(x) sup I f inf I f 0.5 x Figure 5: The oscillation of f over I = [0, 0.5] is osc I (f) = sup I f inf I f Definition Let α R and T = [0, 1] n. spaces on T Then we define the oscillation where V α (T ) := {f : [0, 1] n R measurable with f V α (T ) < }, f V α (T ) := sup ν 0 N ν(α n) Osc(ν, f). We define also the oscillation spaces on R n where V α (R n ) := {f : R n R measurable with f V α (R n ) < }, 17

27 2.3 Lower counterpart classes 2 Definitions f V α (R n ) := sup ν 0 N ν(α n) I =N νn osc I (f). Here the sum is taken over all I as in Definition 2.2.2, where osc I (f) := sup I f inf I f is the oscillation of f over I. Oscillation spaces were considered in [13]. Similarly to Definition 2.1.8, here we define also the classes V α (R n ) := {f : R n R measurable with f V α ε (R n ), ε > 0}, V α+ (R n ) := {f : R n R measurable with f V α+ε (R n ) for some ε > 0}, and the corresponding classes V α (T ) and V α+ (T ). We define also the oscillation class with an exact smoothness α as V α (R n ) := V α (R n ) \ V α+ (R n ), and analogously on T. Remark: (a). V α (T ) and. V α (R n ) are semi-norms in the linear spaces V α (T ) and V α (R n ), respectively. (b) The linear spaces V α (T ) and V α (R n ) are independent (equivalent semi-norms) of N. (c) Consider 0 < α 1, and let f C α (T ), i.e., f : [0, 1] n R with f(x) f(y) c x y α for all x, y [0, 1] n. Then we have the relation f V α (T ). (d) If α > n then the spaces V α (T ) and V α (R n ) contain only the constant functions on T and on R n, respectively. For the case α 0, the spaces V α (T ) (respectively V α (R n )) contain at least all bounded and measurable functions on T (respectively on R n with a compact support). (e) Of course, analogously to remark (e) of Definition 2.1.8, we have the relations (identities and inclusions) δ>α V δ (T ) = V α+ (T ) V α (T ) V α (T ) = δ<α V δ (T ) and the corresponding counterparts for the spaces V α (R n ). (f) A precise characterization of the class V α (R n ) is given by the equivalence f V α (R n ) sup{σ R : f V σ (R n )} = α. 2.3 Lower counterpart classes Below in Sections we will characterize, in terms of upper box dimension, the classes B and V introduced in Definitions and 2.2.4, respectively. The upper 18

28 2.3 Lower counterpart classes 2 Definitions box dimension is given, see Definition 2.4.2, by the lim of a quantity and, as lower counterpart, the lower box dimension is defined in the same way but with the lim instead. Hence, in order to make an approach for the lower box dimension, it seems legitimate to consider and characterize another kind of classes of functions, which are appropriately defined as lower counterparts of those classes B and V. Definition Let α R, 0 < p and s R. We define the classes B s p, (R n ) and V α (R n ) replacing sup by lim in the definitions of semi/quasinorms of the corresponding spaces Bp, (R s n ) and V α (R n ) given, respectively, in Definitions and Thus B s p, (R n ) is the class of all f S (R n ) such that the quantity f B s p, (R n ) := lim j 2 js ϕ j f Lp(R n ) is finite, and V α (R n ) is the class of all f : R n R measurable such that the quantity f V α (R n ) := lim ν N ν(α n) I =N νn osc I (f) is finite. In the same way as in Definition 2.1.8, we define the classes B s p (R n ) := σ<s Bσ p, (R n ), B s+ p (R n ) := σ>s Bσ p, (R n ), B s p (R n ) := B s p (R n ) \ B s+ p (R n ). Similarly, as in Definition 2.2.4, we define the classes V α (R n ) := δ<α V δ (R n ), V α+ (R n ) := δ>α V δ (R n ), V α (R n ) := V α (R n ) \ V α+ (R n ), and analogously on T. Remark: (a) f V α (R n ) and f B s p, (R n ) are not semi/quasi-norms. We call them pseudo-norms. (b) We have a lower counterpart of remark (b) of Definition 2.2.4, i.e., the classes V α (T ) and V α (R n ) are independent (equivalent pseudo-norms) of N. (c) Unlike remark (c) of Definition 2.1.8, here we don t know if the classes B s p, (R n ) are independent, or not, of ϕ and ϕ 0. But this does not matter, since in the present work the point is to compare, in these lower classes, the lower smoothness with the lower box dimension of the graphs. In the next Sections 2.4 and 2.5 we will introduce the necessary concepts about fractal geometry, namely box and Hausdorff dimensions, measure and mass distributions, support of a measure, d-sets and h-sets. Basic properties concerning these concepts 19

29 2.4 Upper and lower box dimensions 2 Definitions are given as well. 2.4 Upper and lower box dimensions Definition Let f : A R n R be a function. Then we define M(ν, f) as the minimum number of N-adic cubes, as in Definition for R n+1, with volume N ν(n+1), required to cover Γ(f) (see Figure 2 above for the case n = 1). Definition (a) Let f : A R n R be a function such that Γ(f) is a non-empty and bounded subset of R n+1. Then we define (see [14], pp. 38 and 41, with elementary adaptations) the upper box dimension of Γ(f), dim B Γ(f) := lim ν log M(ν, f) log N ν. We define also the lower box dimension of Γ(f), dim B Γ(f) := lim ν log M(ν,f) log N ν. Of course, the definition is immediately extendable to dim B E and dim B E, respectively, for a general non-empty bounded set E R n. (b) For a general non-empty set F R n we define dim B F := sup E F dim B E and dim B F := sup E F dim B E, where the supremum is taken over all non-empty bounded sets E F. If the upper box dimension and lower box dimension coincide then we define the box dimension of F, Remark: dim B F, as the common value of them. (a) The definitions of dim B and dim B are independent (same value) of the basis of the logarithms. They are ([14], p. 41) also independent (same value) of the constant N. (b) Consider k N and let f : [ k, k] n R be a bounded function. We have (2k) n N νn M(ν, f) (2k) n N νn N ν 2(sup f + 1), where [ ] represents the integer part of a positive number. This gives n dim B Γ(f) dim B Γ(f) n + 1. (Furthermore, for a general non-empty set F R n, the inequalities dim B F n hold.) 0 dim B F (c) Let f : R n R be a general function. Then, by part (b) of the Definition, we get dim B Γ(f) = sup k N dim B Γ(f [ k,k] n) and the lower counterpart for dim B. 20

30 2.5 Measures, h-sets, and Hausdorff dimension 2 Definitions (d) Let f : R n R be a general function. Then we have the equality dim B Γ(f) = sup k Z n dim B Γ(f k+[0,1] n). (respectively for dim B.) 2.5 Measures, h-sets, and Hausdorff dimension Definition Let = U R n. The diameter of U is U := sup x,y U x y. Remark: We must notice here that when I is a cube as in Definitions 2.2.1/2.2.2 then I still means the volume of I, according to remark (b) of Definition Thus, though in all the work the notation stands for diameter when applied to a general subset of R n, however it means volume when applied to dyadic cubes. We avoid presumable confusions with the notation by making clear the context in each situation. Definition (a) A measure on R n is a function µ : B(R n ) [0, ], defined over all Borel subsets of R n, that satisfies µ( ) = 0, µ(u 1 ) µ(u 2 ) if U 1 U 2, and µ( k N U k) = k N µ(u k) whenever (U k ) k N is a disjoint collection. (b) A mass distribution on R n is a measure on R n that satisfies 0 < µ(r n ) <. (Whenever convenient we will assume, without any loss of generality, that µ(r n ) = 1.) (c) The support of a measure µ is the closed set given by supp µ := R n \ O µ, where the union is taken over all opens sets O µ R n with µ(o µ ) = 0. Definition (see [14], pp ) Consider d 0, δ > 0, and let E be a non-empty and bounded subset of R n. (a) We define H d δ (E) := inf{ k N U k d : U k δ and E k N U k}. (b) The quantity H d δ increases when δ decreases. Hence we can define the d-dimensional Hausdorff measure as H d (E) := lim δ 0 + H d δ (E). (c) There exists a critical value d E 0 such that H d (E) = for d < d E and H d (E) = 0 for d > d E. We define the Hausdorff dimension of E as dim H E := d E. (d) For a general non-empty set F R n, we define dim H F by applying the same extension technique as in Definition (b). 21

31 2.5 Measures, h-sets, and Hausdorff dimension 2 Definitions Remark:(see [14], pp. 43 and 55, with elementary adaptations) (a) If in part (a) of the Definition we impose the same diameter for all U k instead, thus in part (b) we do not obtain necessarily a monotone H d δ on δ, and the function H d supported on E is not necessarily a measure. However, if we take in (b) the upper (respective lower) limit, then in part (c) we obtain the upper (respectively lower) box dimension (cf. Definition 2.4.2); so we have the inequalities 0 dim H F dim B F dim B F n for any non-empty set F R n. Furthermore, if A is a non-empty open subset of R n and f : A R is a general function, then we have also dim H Γ(f) n (besides dim H Γ(f) dim B Γ(f) dim B Γ(f) n + 1). In order to prove that inequality, assume that A is bounded and put D k := {x A : f(x) k}, k N. The Lebesgue measure of D k, λ(d k ), tends (grows) to λ(a) when k, thus λ(d k0 ) > 0 for k 0 N sufficiently large. As dim H Γ(f Dk0 ) dim H D k0 = n, then that inequality follows.) More generally, the inequalities dim(a) dim Γ(f) dim(a) + 1 hold for any set A R n, where dim stands for box or Hausdorff dimensions. (b) Suppose that µ is a mass distribution on R n and B r are open or closed balls with 0 < B r = r < 1 and centered at any P F = supp µ. If r d cµ(b r ) for 0 < r < 1 then dim B Γ(f) d, and if r d cµ(b r ) for 0 < r < 1 then dim H Γ(f) d. In order to avoid any confusion with the notation for balls and respective radii or diameters, namely as far as concerns Definition above on the one hand, and the previous remark (b) and the following Definitions 2.5.5/2.5.6 on the other hand, please recall here the remark of Definition (a). Definition Let Π θ and Ξ θ be two real valued expressions, depending on a parameter θ which runs over all elements of a class given in each context. We say that Π θ and Ξ θ are equivalent, writing Π θ θ Ξ θ, if these two expressions are strictly positive and there exist c 1, c 2 > 0 such that c 1 Ξ θ Π θ c 2 Ξ θ holds for all given θ according to the context. Furthermore, if Π, Ξ R + { }, then Π Ξ means that Π is finite if and only if Ξ 22

32 2.6 Maximal and minimal dimensions 2 Definitions is finite. Definition Consider d > 0 and let E be a non-empty subset of R n. Thus, E is a d-set if there is a mass distribution µ on R n such that µ(b r ) r r d, for all open or closed balls B r with 0 < B r = r < 1 and centered at any P E = supp µ. Remark: (a) If E is a d-set then E is compact in R n ; furthermore, by remark of Definition we have the equalities dim H E = dim B E = dim B E = d. (b) If E is a d-set then, given any ν N 0, all efficient coverings C of E by dyadic cubes have ν,c 2 νd elements. So, dim B (E A) = d + dim B A holds, as well as the corresponding counterpart for dim B, for any A R k, k N. Moreover, by Corollary 7.4 of [14], p. 95, with E in place of F, we have also dim H (E A) = d + dim H A. Definition Here, we generalize Definition Consider h : (0, 1) R + a continuous function satisfying h(0 + ) = 0. By definition, a non-empty set E R n is a h-set if there exists a mass distribution µ on R n such that µ(b r ) r h(r), for all open or closed balls B r with 0 < B r = r < 1 and centered at any P E = supp µ. Remark: Observe that if E is a h-set as defined above, then E is compact in R n. Furthermore, if E is a h-set, then the function h is equivalent (see Definition 2.5.4) to some monotone function h. (Whenever convenient we will consider 0 < r 1, instead of 0 < r < 1, in Definitions 2.5.5/ By the way, the two situations produce equivalent definitions.) 2.6 Maximal and minimal dimensions The Besov class with integrability p and exact smoothness s, B s p (R n ), as well as the classes with smoothness s, Bp s (R n ), and with smoothness s+, Bp s+ (R n ), were introduced in Definition As referred in Section 1.2, one of main aims of the work is to make a characterization, in terms of upper box dimension, of those classes B for a given smoothness. This will be made by calculating maximal and minimal dimensions, 23

33 2.6 Maximal and minimal dimensions 2 Definitions according to the following Definition. Definition Let 0 < p and s > 0. The upper box dimension dim B was introduced in Definition Then we can define the maximal dimension Dim s p := sup dim B Γ(f), f Bp s (R n ) where the supremum is taken over all continuous real functions f B s p (R n ). We define also the minimal dimension dim s p := inf f B s+ p (R n ) dim BΓ(f), the infimum taken over all continuous real functions f Bp s+ (R n ) with a compact support. Remark: (a) These definitions seem different from the corresponding ones given in the Section 1.2, and are more appropriate to developing research in this topic. Nevertheless, as we will see in Remark (b), these apparently different notions coincide. (b) If P is a non-zero real polynomial, then P [0,1] n +k C ([0, 1] n +k) C 1 ([0, 1] n +k) for all k Z n. Therefore we have dim B Γ(P ) = n. This follows by remark (d) of Definition 2.4.2, remark (c) of Definition 2.2.4, and Theorem below. Nevertheless, by the property F (Φ Ψ) = (2π) n 2 (F Φ)(F Ψ) on S (R n ), we have P B s p,q (R n ) = (2π) n 2 F 1 ((F ϕ 0 )(F P )) Lp (R n ) = (2π) n 2 P Lp (R n ) =, and therefore P B s p,q(r n ), so P B s+ p (R n ), for any 0 < p < and 0 < q and any s R. Moreover, given any s R and 0 < p <, there is a bounded and continuous function g s,p : R n R satisfying g s,p C ([0, 1] n + k) for all k Z n, so dim B Γ(g s,p ) = n, such that the relation g s,p B s p (R n ) holds. This follows by considering g s,p (x) := g s (x 1,..., x n ) := j 1 2 js ψ (x 2je 1 ) cos(2 j x 1 ) if s 0, respectively g s,p (x) := g s,p (x 1,..., x n ) := j 1 ψ ( 2 js p n (x 2η j e 1 ) ) cos(2 j x 1 ) if s 0, where e 1 := (1, 0,..., 0) R n, η j := j j =1 2j s p n, and ψ S(R n ) \ {0} is real and with supp ψ B 1 (0), and by using a direct generalization of Theorem 4.2.1, a trigonometric counterpart of Corollary 9 of [37], p (Of course, we can also prove this assertion in the basis of Corollary 9 of [37] with an appropriate wavelet expansion g s,p, instead of a trigonometric one.) (As a matter of fact, we can even state the following: Given any s R and any 0 < p < 24

34 3 Dimensions and smoothness in function spaces, then there is a (bounded and continuous) real function g s,p C (R n ) such that the relation g s,p Bp s (R n ) holds: take for instance g s,p (x) := ( ) j 1 2 2j ψ 2 (js 2j ) p n x cos(2 j x 1 ), where ψ S(R n ) \ {0} is real.) Hence, in order to enrich the definition of minimal dimension for 0 < p < and s > 0, we introduced above the compact support condition. (Otherwise, the minimal dimension would be n for any of those parameters p and s, as follows by using the functions P or g s,p as in the comments above.) Nevertheless, in some situations (see comments before Theorem 3.6.3) the class of all continuous real functions with a compact support is not appropriate indeed, so we consider the somewhat larger and convenient class of all bounded and continuous real functions. However, many of the results presented in this work do no longer hold for this larger class of functions. (c) We have dim s p Dim s p for 0 < p and s > 0. In fact, by Theorem 1.1.1, the class Bp s (R n ) = Bp s (R n ) \ B s+ (R n ) contains a non-empty subset of continuous real functions with a compact support. p (d) Of course, analogous definitions for maximal and minimal dimensions make sense for lower box and Hausdorff dimensions, as well as when we consider the lower counterpart classes according to Definition Moreover, the previous part (c) remains true for those other dimensions and classes (cf. Theorem below). 3 Dimensions and smoothness in function spaces 3.1 Summary of the Chapter Consider an integrability 0 < p and an exact smoothness s > 0. Hence, for a given class Bp s (R n ), represented by a point in the ( 1, s)-plane, we will determine p maximal and minimal values for upper and lower box dimensions, for the graphs of all continuous real functions with a compact support and represented by this point. In order to achieve this main aim, firstly in Section 3.2 we point out the intimate 25

35 3.2 Box dimensions in oscillation spaces 3 Dim. Smooth. F.S. conection between box dimensions and oscillation spaces and then, in Section 3.3, by developing embeddings or inclusions between oscillation spaces and Besov spaces, we get estimates for maximal and minimal values for the box dimensions of the graphs. Secondly, in Section 3.4 we calculate the box dimension for graphs of some bizarre specially constructed functions, including Weierstrass-type and chirp-type ones, in order to prove that the estimates of the previous Section are actually the maximal and minimal values of the box dimensions, and then in Section 3.5 we state the main result, where we give two tables summarizing maximal and minimal box and Hausdorff dimensions (in what concerns Hausdorff dimension, we refer to Roueff thesis [25] and even to [26]). When we search for minimal lower box dimension, we pay special attention to scalesparse Weierstrass-type functions, namely in Theorem As we will see later in Section 4.2, this simple and easy result will play an important role in what concerns the search for graphs which are d-sets. In Section 3.6 we deal again with the relation between smoothness and dimensions of the graphs, but now from a quite different point of view, which consists in applying the lifting operator to a fixed function, ranging in this way throuth the smoothness, and observing the consequente translation in terms of box and Hausdorff dimensions. Here we obtain additional and distint information, in particular we conclude that there is some uncertainty in that relation between smoothness and dimensions. 3.2 Box dimensions in oscillation spaces We start by a precise characterization of the oscillation spaces V α (R n ) and corresponding lower counterpart classes V α (R n ), in terms of upper and lower box dimensions, respectively. Afterwards, in Section 3.3, these spaces and classes will be related with Besov spaces and respective lower counterparts, since as we have seen in Section 1.1 both parameters α and s measure smoothness. Theorem Let f : [0, 1] n R be a continuous function. Then we have the equivalences 26

36 3.2 Box dimensions in oscillation spaces 3 Dim. Smooth. F.S. (a) dim B Γ(f) n + 1 γ f V γ (T ), in the case 0 < γ 1, (b) dim B Γ(f) n + 1 γ f V γ+ (T ), in the case 0 γ < 1. Remark (a) The Theorem is also true for dim B, V γ (T ), V γ+ (T ) in place of the standard counterparts. (b) The Theorem and the previous part (a) are immediately extendable to V γ (R n ), V γ+ (R n ) (respectively V γ (R n ), V γ+ (R n )) for continuous functions f : R n R with a compact support. (c) Consider f : R n R as a general measurable function. Then for 0 < γ 1 we have the implication dim B Γ(f) n + 1 γ = f V γ (R n ), and for 0 γ < 1 the implication dim B Γ(f) n + 1 γ = f V γ+ (R n ). We have also the lower counterparts for dim B, V γ (R n ), V γ+ (R n ). We should notice that in Theorem 3.2.1, as f : [0, 1] n R is continuous, then both sides of first equivalence are true if γ = 0, and the left hand side of second equivalence is true if γ = 1. This follows by remark (d) of Definition and remark (b) of Definition As a natural consequence of the previous results we get the following Theorem, which will be convenient when we intend to deal both with supports and dimensions of functions. Definition Let = A R and consider f : A C. Then f is a Lipschitz function if f(x 1 ) f(x 2 ) c f x 1 x 2, x 1, x 2 A. (We call c f a Lipschitz constant of f.) Theorem Let f : R n R be a continuous function, and let η : R n R be a Lipschitz function. Then we have (cf. remark of Proposition below) (a) dim B Γ(ηf) dim B Γ(f), (b) dim B Γ(ηf) dim B Γ(f). 27

37 3.3 Embeddings between Besov and oscillation spaces 3 Dim. Smooth. F.S. 3.3 Embeddings between Besov and oscillation spaces Now, by developing inclusions or embeddings, we will relate the oscillation spaces V α (R n ) with the Besov spaces B s p,q(r n ). In papers [17] and [18] we find very fine embeddings concerning those spaces. Nevertheless, here we make a somewhat different approach to oscillation spaces, since we defined them in Section 2.2 by taking into account only first order differences, instead of greater order ones. In what follows, if (S 1,. S1 ) and (S 2,. S2 ) are two semi/quasi-normed spaces of tempered distributions then, by definition, the intersection of the two spaces must be understood as the semi/quasi-normed space S 1 S 2, provided by definition with the semi/quasi-norm. S1 S 2 :=. S1 +. S2. Furthermore, let us temporarily consider V A,α (R n ) and B A,s p,q (R n ) as the restrictions, of the spaces V α (R n ) and B s p,q(r n ), to the measurable real functions with support contained in a fixed non-empty bounded set A R n. Consider 0 < p < and γ n. Then the Theorem below gives the embedding p B A,γ p,1 (R n ) V A,min{1,γ} (R n ). We require this embedding in order to prove the Corollary 3.3.2, that gives an upper bound for Dim s p when 0 < p and γ > n p. On the other hand, the Theorem gives a converse result. Considering 0 < p < and γ R and taking into account the definitions given above, we have the embedding L max{1,p} (R n ) V A,γ max{1,p} (R n ) B A,γ p, (R n ). As a consequence this embedding originates the Corollary 3.3.5, that generalizes Theorem and gives a lower bound for dim s p when 0 < p < and 0 < s < 1 max{1,p}. Theorem Consider 0 < p < and γ n p, and let f : Rn R be a function such that f B γ p,1(r n ) and supp f B R (0), where R > 0. Then, it holds f V min{1,γ} (R n ) c R f B γ p,1 (Rn ). 28

38 3.3 Embeddings between Besov and oscillation spaces 3 Dim. Smooth. F.S. Corollary Consider 0 < p and γ > n p, and let f : Rn R be a function such that f Bp γ (R n ). Then we have (see Figure 6) the inequality dim B Γ(f) n + 1 min{1, γ}. In [29], p. 21, we find a result very related with the Corollary above. By the way, for a comparison between [29], p. 21, and Corollary 3.3.2, see for instance [31], pp. 189 and 192 also Lemma when 0 < p < 1. For detailed calculations concerning the case p =, see e.g. [33], p Remark (a) At least with the further assumption 0 < γ < 1, the Theorem and Corollary are sharp results, in the sense that we cannot replace the expression min{1, γ} by a strictly larger number. This follows by Theorem (b) Consider p, γ and f as in Corollary Suppose additionally that γ 1. Then by remark (a) of Definition we have dim H Γ(f) = dim B Γ(f) = n. A deeper result concerning Hausdorff dimension will be obtained below in Theorem (c) As an extension or special case of Theorem 3.3.1, we have also the embedding (the prefix Re means that we must consider the restriction to the real functions) ReB n 1,1(R n ) V 1 (R n ). (d) We have no lower counterpart of Corollary for B γ p (R n ) and dim B in place of the standard/upper counterparts. This follows by Remark (c). 29

39 3.3 Embeddings between Besov and oscillation spaces 3 Dim. Smooth. F.S. s 1 Dim s p n n + 1 s s = n/p Figure 6: The Corollary gives Dim s p n + 1 min{1, s} 1/n 1/p for 0 < p and s > n p Theorem Consider 0 < p < and γ R, and let f V γ max{1,p} (R n ) be a (measurable) function such that supp f B R (0), where R > 0. Then, it holds f B γ p, (R n ) c R f Lmax{1,p} (R n ) V γ max{1,p} (R n ). Corollary Consider 0 < p < and 0 γ < 1 max{1,p}, and let f : Rn R be a continuous function with a compact support. Suppose additionally that f B γ+ p (R n ). Then by Theorem and Remark (b) we have (see Figure 7) the inequality dim B Γ(f) n + 1 γ max{1, p}. 1 Remark (a) At least for 0 < γ <, the Theorem and Corollary max{1,p} are sharp results, in the sense that we cannot replace the expression γ max{1, p} by a strictly smaller number. This follows by Theorem (b) We have also a kind of lower counterpart inequality for the inequality of Theorem Actually, with the same proof we can show that the inequality f B γ p, (R n ) c R f V γ max{1,p} (R n ) holds. Hence, by Remark (b) it follows that the Corollary is valid also for B γ+ p (R n ) and dim B in place of the standard/upper counterparts. 30

40 3.3 Embeddings between Besov and oscillation spaces 3 Dim. Smooth. F.S. (c) Consider 1 p < and γ R. As can be seen in the Proof of Theorem 3.3.4, under these range of parameters the constant c R of this Theorem does not depend on R; moreover, the embedding L p (R n ) V γp (R n ) Bp, (R γ n ) holds. Furthermore, as will be given in Theorem below, the Proof of which takes advantage of Young s inequality see Lemma 6.3.4, for 1 p < and γ R it holds also the embedding L 1 (R n ) V γp (R n ) Bp, (R γ n ). Of course, for γ = 1 max{1,p} the Corollary is also valid. This follows by remark (b) of Definition We must recall here the comments given before Theorem 1.2.1, which follow by Theorem and are explained in comments following it based in Remark According to this, it turns out also that the Corollary cannot be improved with the other inequality. s 1 dim s p s = 1/p s = 1 n + 1 sp 1 n + 1 s 1/p Figure 7: The Corollary gives dim s p n + 1 s max{1, p} for 0 < p < and 0 < s < 1 max{1,p} 31

41 3.4 Functions for extremal box dimensions 3 Dim. Smooth. F.S. 3.4 Functions for extremal box dimensions Next we will construct extreme functions with the goal of reaching the maximal and minimal dimensions. In Definition we construct a function that materializes the maximal dimension for 0 < p < and 0 < s n, as pointed out by Theorem p Furthermore, in Definition we have a more refined function in order to get the Theorem 3.4.4, which generalizes Theorem as far as concerns exact smoothness. On the other hand, in Definition we have a function Λ that constitutes an approach in order to achieve the minimal dimension for 1 p < and 0 < s 1 p. Definition Let ψ : R n R with ψ S(R n ), ψ(0) 0 and supp(f ψ) {ξ R n : 1 a ξ 1 + a}, for a > 0 sufficiently small. Let j 0 N 0 be sufficiently large. (I) For n = 1: For each j N \ {1, 2} let Θ j : R R with Θ j := ψ ( 2 j+j 0 ( 1/ ln j) ). Let Θ : R R with Θ := j 3 Θ j. (Pointwise convergence and also convergence in S (R) with the strong topology to the same function.) Let φ : R R, with φ C 0 (R) and satisfying φ(x) = x, x [0, 1]. Let Θ 0 : R R with Θ 0 := φ Θ. (II) For n N \ {1}: For each j N \ {1, 2} and for l {1,..., j 2(n 1) } let Θ j,l : R n R with Θ j,l := ψ ( 2 j+j 0 ( m(j, l)) ), ( 1 and with m(j, l) :=, s 2,l,..., s n,l ), where the j 2(n 1) different values of l correspond ln j j 2 j 2 to the different values of (s 2,l,..., s n,l ) {1,..., j 2 } n 1. Let Θ : R n R with Θ := j 2(n 1) j 3 l=1 Θ j,l. (Pointwise convergence and also convergence in S (R n ) with the strong topology to the same function.) 32

42 3.4 Functions for extremal box dimensions 3 Dim. Smooth. F.S. Let φ : R n R, with φ C 0 (R n ) and satisfying φ(x) = φ(x 1,..., x n ) = x 1, x [0, 1] n. Let Θ 0 : R n R with Θ 0 := φ Θ. Theorem Let Θ 0 be as in Definition Then (see Figure 8) (a) Θ 0 is continuous, (b) dim B Γ(Θ 0 ) = n + 1, (c) Θ 0 B n p p (R n ) for 0 < p. s Dim s p 1 s = n/p n + 1 n + 1 Figure 8: The Theorem gives Dim s p = n + 1 for 0 < p < 1/n 1/p and 0 < s n p As mentioned in Section 1.2, the Theorem cannot be improved with the inequality. In the same way, only the part of the claimed Th. 4.2 of [13], p. 220, is true. This follows by Theorem (or by Theorem below). The failure of part of this claim in [13], p. 220, comes from the false inequality (4.3) of [13], p. 219, which originates the false inequality (4.1) of [13], p We can see in Remark below that (4.1) f V γ (R) c f B γ 1,1 (R) with 0 < γ < 1 of [13], p. 219, is false, even with the restriction to the real functions f belonging to C 0 (R). In [20], Kamont and Wolnik 33

43 3.4 Functions for extremal box dimensions 3 Dim. Smooth. F.S. also constructed a counterexample for this claim of [13], p Lemma ([31], p. 195) Let ϕ C 0 (R n ). Let 0 < p 0 p 1, 0 < q and s R. Then f ϕf yields a continuous linear mapping from B s p 1,q(R n ) into B s p 0,q(R n ). Remark Let ϕ be as in Definition 2.1.8, and with supp(f ϕ) {ξ R n : 1(1 + a) ξ 2(1 a)} and (F ϕ)(ξ) = 1 if 1 a ξ 1 + a, for some a > 0. 2 Let ψ : R n R with ψ S(R n ) \ {0} and supp(f ψ) {ξ R : 1 a ξ 1 + a}. Furthermore, let φ : R n R with φ C 0 (R n ) and suppose that (φψ)(0) 0. For each j N let ψ j := ψ(2 j ) and φ j := φψ j ; thus we have φ j C 0 (R n ). Then, for α R it holds φ j V α (R n ) φ j L (R n ) φ j (0) = c > 0. Nevertheless, from the well-known property F (Φ Ψ) = (2π) n 2 (F Φ)(F Ψ) on S (R n ), it holds (besides ϕ j ψ j = 0 for j N 0 and j j) ϕ j ψ j = (2π) n 2 ψ j, if in Definition we consider ϕ as above. So the identities ψ j B s p,q (R n ) = 2 js (2π) n 2 ψ j Lp (R n ) = c2 j(s n p ) hold for 0 < p, 0 < q and s R. Hence, by Lemma we obtain the estimation φ j B s p,q (R n ) c 2 j(s n p ). Definition Let ψ : R n R with ψ S(R n ) \ {0} and supp(f ψ) {ξ R n : 1 a ξ 1 + a}, for a > 0 sufficiently small. Let 0 < λ 1. For each j N \ {1, 2} define k(j) := [2 j(1 λ) ]. (I) For n = 1: For each j N \ {1, 2} and for k {1,..., k(j)} let m(j, k) := 1 ln j k2 j and Θ j,k : R R with Let Θ : R R with Θ := j 3 Θ j,k := ψ ( 2 j ( m(j, k)) ). k(j) k=1 Θ j,k. (Pointwise convergence and also convergence in S (R) with the strong topology to the same function.) Let φ : R R, with φ S(R) and satisfying φ(x) = x, x [0, 1]. 34

44 3.4 Functions for extremal box dimensions 3 Dim. Smooth. F.S. Let Θ 0 : R R with Θ 0 := φ Θ. (II) For n N \ {1}: Let 0 h n 1. For each j N \ {1, 2} define d(j) := d(j) := j 2 if h = n 1.) ] [2 j n 1 h n 1. (Modification For each j N \ {1, 2}, for k {1,..., k(j)} and for l {1,..., d(j) n 1 } let m(j, k, l) := ( 1 ln j k2 j, s 2,l d(j),..., s n,l d(j) ), where for each fixed (j, k) the d(j)n 1 different values of l correspond to the different values of (s 2,l,..., s n,l ) {1,..., d(j)} n 1. Let also Θ j,k,l : R R with Let Θ : R n R with Θ := j 3 Θ j,k,l := ψ ( 2 j ( m(j, k, l)) ). k(j) d(j) n 1 k=1 l=1 Θ j,k,l. (Pointwise convergence and also convergence in S (R n ) with the strong topology to the same function.) Let φ : R n R, with φ S(R n ) and satisfying φ(x) = φ(x 1,..., x n ) = x 1, x [0, 1] n. Let Θ 0 : R n R with Θ 0 := φ Θ. Theorem Let 0 h n 1, 0 < λ 1 and Θ 0 as in Definition Then (a) Θ 0 is continuous, (b) dim B Γ(Θ 0 ) = n + 1, (c) Θ 0 B h+λ p p (R n ) for 0 < p. The Theorem strengthens and generalizes Theorem In fact, if in Definition we additionally impose ψ(0) 0, φ C 0 (R n ), replace ψ by ψ(2 j 0 ), take h = n 1, λ = 1 and simplify 1 ln j 2 j by 1 ln j, then we get the same Θ and Θ 0 of Definition In the following Definition 3.4.3, the function Λ constitutes an approach in order to achieve the minimal dimension for 1 p < and 0 < s 1. The Theorem p deals with the product φλ, in order to be able to guarantee a compact support which in principle is not guaranteed in Definition

45 3.4 Functions for extremal box dimensions 3 Dim. Smooth. F.S. We can realize here that the representation in wavelets, or even in quarks in the context of [34], are possible alternatives to Definitions 3.4.1, and 3.4.3, specially for the later one, in order to construct extremal functions. The former approach is given in [18], dealing with a multi-fractal analysis on wavelet expansions. For such an approach, an useful tool to manage the smoothness is the Theorem in p. 194 of [37]. Definition Let ψ : R n R with ψ S(R n ) \ {0} and supp(f ψ) {ξ R n : 1 a ξ 1 + a}, for a > 0 sufficiently small. Let 0 < λ 1. For each j N define k(j) := [(1 2 λ )2 j(1 λ) ]. In what follows, for λ = 1 we put k(j) = 1 and replace 2 jλ k2 j by 0 and 1 j by 1 j 2. (I) For n = 1: For each j N and for k {1,..., k(j)} let m(j, k) := 2 jλ k2 j and Λ j,k : R R with Let Λ : R R with Λ := j 1 Λ j,k := 1 j ψ ( 2 j ( m(j, k)) ). k(j) k=1 Λ j,k. (Uniform convergence.) (II) For n N \ {1}: For each j N, for k {1,..., k(j)} and for l {1,..., 2 j(n 1) } let m(j, k, l) := ( 2 jλ k2 j, s 2,l,..., s ) n,l 2 j 2, where for each fixed (j, k) the 2 j(n 1) different values of l j correspond to the different values of (s 2,l,..., s n,l ) {1,..., 2 j } n 1. Let also Λ j,k,l : R R with Λ j,k,l := 1 j ψ ( 2 j ( m(j, k, l)) ). Let Λ : R n R with Λ := j 1 k(j) 2 j(n 1) k=1 l=1 Λ j,k,l. (Uniform convergence.) Theorem Let 0 < λ 1 and Λ as in Definition Then (a) Λ is bounded and continuous, (b) dim B Γ(Λ) n + 1 λ, 36

46 3.5 Summarizing maximal and minimal dimensions 3 Dim. Smooth. F.S. (c) Λ B λ p p (R n ) for 0 < p. Actually, we can write the identity dim B Γ(Λ) = dim B Γ(φΛ) = n + 1 λ, where φ is an appropriate infinitely smooth and real cutoff function. This follows by the following Theorem, together with respective Proof, which is an improvement of the previous one. Theorem Consider 0 < λ 1 and Λ as in Definition 3.4.3, and let φ : R n R, with φ S(R n ) and satisfying φ(x) = 1, x [0, 1] n. Then we have the relations (a) dim B Γ(φΛ) = n + 1 λ, (b) φλ B λ p p (R n ) for 0 < p. 3.5 Summarizing maximal and minimal dimensions We are now in a position to give the exact values for maximal and minimal upper box dimensions, as well as the maximal lower box dimension, over all continuous real functions compactly supported on R n, with integrability 0 < p and exact smoothness s > 0. The following Theorem and Remark make exactly this. Theorem Let 0 < p and s > 0. Then we have (see Figures 9, 10, 11) (a) Dim s p = n + 1 if 0 < s n p, (b) Dim s p = n + 1 min{1, s} if s > n p, and (c) dim s p = n + 1 s max{1, p} if 0 < s 1 max{1,p}, (d) dim s p = n if s 1 max{1,p}. Remark (a) As we can see in the arguments of the Proof of Theorem 3.5.1, the supremum Dim s p is a maximum and the infimum dim s p is a minimum. (b) If when taking the supremum and the infimum in Definition we suppose 37

47 3.5 Summarizing maximal and minimal dimensions 3 Dim. Smooth. F.S. f B s p (R n ), then we have the same results of Theorem for Dim s p and dim s p, including part (a) of present Remark. In fact, as we can see in the Proof of the Theorem, we need only replace Theorem by the more general Theorem (c) We define Dim s p if when taking the supremum in Definition we replace dim B by dim B. As can be seen in the Proof of Theorem 3.5.1, we have Dim s p = Dim s p for 0 < p and s > 0; moreover, the previous parts (a) and (b) of the present Remark remain true for Dim s p. (d) Looking at the Proof, we realize that for obtaining parts (a), (b) and (d) of Theorem 3.5.1, we didn t need at all to impose any continuity condition in Definition 2.6.1; and this is true also for the previous parts of the present Remark. (In fact, by recalling [31], p. 131, we know that in the case s > n p as in part (b) of the Theorem we have only continuous functions; and concerning part (d) and the case 0 < s n p as in part (a), we know by remark (a) of Definition that the upper box dimension falls between n and n + 1.) On the other hand, if in Definition we relax the continuity on a subset of R n of upper box dimension n 1 (for example on a hyperplane), then even the part (c) of Theorem remains true, including the previous parts of the present Remark. This will be proved in Section 5.4. (e) Concerning the extension of the present Remark to functions which are not compactly supported, see at the end of remark (b) of Definition

48 3.5 Summarizing maximal and minimal dimensions 3 Dim. Smooth. F.S. s 1 Dim s p n s = n/p n + 1 s n + 1 n + 1 Figure 9: The Theorem gives the exact values of Dim s p for 0 < p and s > 0 1/n 1/p s 1 dim s p n n s = 1/p s = 1 n n + 1 sp 1 n + 1 s 1/p Figure 10: The Theorem gives the exact values of dim s p for 0 < p and s > 0 We also present an alternative graphical representation, to facilitate the interpretation. 39

49 3.5 Summarizing maximal and minimal dimensions 3 Dim. Smooth. F.S. dim(graphf) n+1 Figure 11: Case n p < 1. The solid line represents Dim s p and the dashed line represents dim s p. For the n case n p is simpler > 1 the solid line 0 1 max(1,p) n p 1 s We gave, in Theorem and Remark above, the maximal and minimal values of the upper box dimension and the maximal value of the lower box dimension, for an integrability 0 < p and an exact smoothness s > 0. So, the Corollary below complements these results, by giving the minimal value for the lower box dimension. On the other hand, Roueff in [25] gives the maximal and minimal Hausdorff dimensions. The following Theorem has an easy and short Proof. It is an extension of the corresponding result in [25], 2.5.4, p. 29, is also true in the case γ = and is crucial in order to prove the Corollary Theorem Let ζ : R R be a bounded and Lipschitz function. Consider (θ j ) j N R, (a j ) j N R with a j 2 λ js, and (λ j ) j N R + satisfying λ j+1 λ j c > 0. Define γ := lim j λ j+1 λ j (hence γ [1, ] holds). For 0 < s 1, let W s : [0, 1] R be defined by W s (x) := j 1 a jζ(2 λ j x + θ j ). Then we have dim B Γ(W s ) 2 Ξ(γ, s), γs where Ξ(γ, s) := 1 s+γs. We observe the following: For γ = we have Ξ(, s) = 1 whenever 0 < s 1; for (finite) fixed γ 1, the one variable function Ξ(γ, ) is strictly increasing and ranges over (0, 1] for 0 < s 1; analogously, for 40

50 3.5 Summarizing maximal and minimal dimensions 3 Dim. Smooth. F.S. fixed 0 < s 1, the function Ξ(, s) is strictly increasing and ranges over [s, 1] for 1 γ. Of course, with 0 < s < 1 the quantity 2 Ξ(γ, s) = s 1 s+sγ is a number strictly between 1 and 2. Below, in Theorem and comments that follow it, we will see that in many cases the inequality in the Theorem above is actually an equality. Corollary Let ψ S(R n ) \ {0} be a real function. Consider s > 0 and let W s : R n R be defined by W s (x 1,..., x n ) := j 1 2 j!s cos(2 j! x 1 ), where j! = j(j 1) Then we have the relations dim B Γ(ψW s ) = n and ψw s B s p (R n ) for 0 < p. Proof. The Theorem gives the second relation. On the other hand, by remark (b) of Definition and by Theorem with γ =, we obtain n dim B Γ(ψW s ) dim B Γ(W s ) (2 1) + (n 1), where the second (middle) inequality is justified by Theorem (b) with elementary adaptations. In particular the Corollary shows that, for 0 < p and s > 0, there is a continuous real function h s with a compact support on R n, an integrability p and an exact smoothness s (taking h s := ψw s ), that achieves the minimal lower box dimension n. Of course, by remark (a) of Definition this function achieves also the minimal Hausdorff dimension dim H Γ(h s ) = n, which is given also by Lemme 1 of [25], p. 13. Concerning non-linearity of lower box and Hausdorff dimensions, which is a closely related topic, see [23, 25, 38]. The following Theorem is the heart of the Sections Summary of maximal and minimal dimensions: Theorem Let 0 < p and s > 0, and define ( ), ( ) and ( ) as follows: ( ) := n + 1 if 0 < s n p and ( ) := n + 1 min{1, s} if s > n p ; ( ) := n + 1 s max{1, p} if 0 < s 1 max{1,p} and ( ) := n if s 1 max{1,p} ; 41

51 3.6 Lifting operator and dimensions 3 Dim. Smooth. F.S. ( ) := n + 1 s min{1, p} if 0 < s 1 min{1,p} and ( ) := n if s 1 min{1,p}. Then, the following tables hold. The first table contains maximal and minimal dimensions over B s p (R n ), the second table does the same over B s p (R n ). Maximal Minimal dim B ( ) ( ) dim B ( ) n dim H ( ) n Maximal Minimal dim B n + 1 ( ) dim B n + 1 ( ) dim H n + 1 n 3.6 Lifting operator and dimensions In Theorem we dealt with the interval ranged by dimensions of the graphs of continuous real functions, for a given smoothness between 0 and 1. In fact, this Theorem shows already some uncertainty in the relation between smoothness and dimensions. Now, we want to investigate the behavior of the fractal dimensions on the graph when we apply a variation of smoothness on a fixed continuous real function. Indeed, as is confirmed by Lemma 3.6.2, the lifting operator is a convenient tool when checking the behavior of dimensions against smoothness. We are not interested in the typical behavior, instead we consider representative extremal cases in order to prove the key result of this Section, Theorem 3.6.3, which summarizes some remarkable consequences concerning extremal behaviors in the relation between dimensions and smoothness. Firstly, however, we need the following two Lemmas in which the Proof of the Theorem is based on. Lemma Consider 1 < d 1 < d 2 < 2 fixed, and let Ξ : (1, ) (0, 1) (0, 1) γs be defined by Ξ(γ, s) := 1 s+γs, for all γ > 1 and all 0 < s < 1, as in Theorems 3.5.3/ Let b 1 := 2 d 1 and b 2 := 2 d 2, thus 0 < b 2 < b 1 < 1. For this fixed pair (b 1, b 2 ), we will consider (s, t) = (s, t)(γ) as a function of γ > 1, defined by the two identities 2 Ξ(γ, s) = 2 b 2 = d 2 and 2 Ξ(γ, s + t) = 2 b 1 = d 1. In particular, 42

52 3.6 Lifting operator and dimensions 3 Dim. Smooth. F.S. we define in this way the function t : (1, ) (0, 1) by t = t(γ) satisfying those two identities. Then, we have the following properties: (a) The function t(γ) satisfies t(1 + ) := lim γ 1+ t(γ) = b 1 b 2 and lim γ t(γ) = 0. (b) If b 1 + b 2 1 (= b 2 1/2 d 2 3/2), i.e. d 1 + d 2 3, then t(γ) is strictly decreasing on γ (1, ), consequently the supremum of t(γ) is given by t sup := t(1 + ) = b 1 b 2 = d 2 d 1. In this case, t(γ) ranges over the interval (0, t sup ). (b ) If b 1 + b 2 > 1, i.e. d 1 + d 2 < 3, then t(γ) is strictly increasing on γ (1, γ 0 ) and strictly decreasing on γ (γ 0, ), where γ 0 := is strictly greater than 1 b 1 b 2 (1 b 1 )(1 b 2 ) in this case; consequently the supremum of t(γ) is a maximum, and is given by t max := t(γ 0 ) = (b 1 b 2 ) b 1 b 2 (1 b 1 )(1 b 2 ) b 1 b 1 b 2 + b 1 b 2 (1 b 1 )(1 b 2 ) b 2 b 1 b 2 + b 1 b 2 (1 b 1 )(1 b 2 ). In this case t max is strictly greater than t(1 + ) = b 1 b 2, and t(γ) ranges over (0, t max ]. (c) Finally, we observe that for fixed d 2, i.e. b 2 fixed, then t max is strictly increasing on b 1, and moreover we have the equalities lim d1 1 + t max = lim b1 1 t max = 1. On the other hand, for fixed b 1 then t max strictly increases when b 2 decreases, and furthermore we have the equalities lim d2 2 t sup = lim b2 0 + t sup = lim b2 0 +(b 1 b 2 ) = b 1. Lemma ([31], pp ) Consider 0 < p, 0 < q, s R and σ R, and let I σ be the lifting operator as in Definition Then, I σ maps B s p,q(r n ) isomorphically onto B s σ p,q (R n ). (This shows explicitly why the lifting operator is the adequate tool to be used in order to apply a variation of smoothness.) The Theorem below gives an immediate application of Theorem and Lemma Before stating that assertion, we should remind that although we are mainly interested in continuous real functions with a compact support, this class in not appropriate (is too small) when we deal with the lifting operator, so we consider here the somewhat larger (and convenient) class of all bounded and continuous real functions. (Concerning details and reasons why we restrict us to these classes of continuous real functions, see the comments below, as well as the remark (b) of Definition ) 43

53 3.6 Lifting operator and dimensions 3 Dim. Smooth. F.S. For functions belonging to such a class, we have the following property: First we must have f B, (R 0 n ). (Cf. comments before Theorem ) Secondly, if t 1 then by Lemma we obtain I t f B, (R 1 n ), so by Theorem we have dim Γ(I t f) = n for both box and Hausdorff dimensions. So, though the dimension of the graph of f can assume any value in the interesting interval [n, n+1], the dimension of the one of I t f will always assume the value n. This is the reason why we will restrict us to 0 < t < 1 in the following assertion. Theorem Consider 0 < t < 1 and let I σ be the lifting operator as in Definition Then, for n d 1 < d 2 n + 1 (also d 1 > n in part (b)) the following holds: (a) If 0 < t d 2 d 1, there is a bounded and continuous real function f such that dim B Γ(I t f) = dim B Γ(I t f) = d 1 and dim B Γ(f) = dim B Γ(f) = d 2. (b) In the case d 1 + d 2 2n + 1 we assume that 0 < t t sup = d 2 d 1, see Lemma (b). However, in the case d 1 + d 2 < 2n + 1 we assume that 0 < t t max, where t max is given as in Lemma (b ) for b 1 := n + 1 d 1 and b 2 := n + 1 d 2. (Notice, see Lemma (c), that lim d1 n + t max = 1 and lim d2 n+1 t sup = n+1 d 1.) Then, for this (d 1, d 2, t), there is a bounded and continuous real function f such that dim H Γ(I t f) = dim B Γ(I t f) = d 1 and dim H Γ(f) = dim B Γ(f) = d 2. (In both parts (a) and (b) the obtained function I t f, by lifting f, is also continuous.) This Theorem clearly shows uncertainty in the relation between dimensions and smoothness, which we already saw in Theorem from a quite different point of view. In fact, the Theorem above tell us that, for any prescribed (t, d 1, d 2 ) fixed and satisfying some appropriate hypotheses, we find a function f t,d1,d 2 ( ) in which an increase of t in the smoothness, by applying the lifting operator, causes a decrease of d 2 d 1 in the dimension, contrasting with Theorem below, where the relation dimensions-smoothness is well defined and very simple for almost all Weierstrass functions. It arises now the Open problem : Considering t > 0, we question if dim Γ(f) 44

54 4 Fractal geometry of Weierstrass-type functions dim Γ(I t f) holds whenever f and I t f are continuous real functions, where dim stands for box or Hausdorff dimensions. 4 Fractal geometry of Weierstrass-type functions 4.1 Summary of the Chapter We started by dealing with box dimensions throughout Sections , including also Hausdorff dimension in Section 3.6. In the next Sections we will consider both, but giving increasingly more emphasis on the Hausdorff dimension. Afterwards we focus on graphs which are d-sets and even h-sets, ending with graphs of functions defined themselves on h-sets. In Sections 4.2, 4.4 and 4.5 we will deal with a kind of functions characterized by a good uniformity of the fractal structure over all the domain, as occurs for the well-known Weierstrass function. These functions illustrate very well the intimate connection between smoothness, on the one hand, and the fractal geometry on the other hand (cf. Section 1 of [12]). The nomenclature Weierstrass function, as well as Weierstrass-type function, is being used somewhat loosely in the present work. In Section 4.2 we estimate the smoothness and box and Hausdorff dimensions of some trigonometric Weierstrass-type functions, including the Hausdorff dimension in the scale-sparse case. In Section 4.3 we point out some more or less striking differences between box and Hausdorff dimensions, when applied to graphs of functions, as well as some situations when it is convenient to deal with one of them and not the other. In Section 4.4 we investigate some criteria in order to decide when the graph of a function is not a d-set. Moreover, for each d between n and n + 1 we construct a function on [0, 1] n the graph of which is a d-set. In Section 4.5, for the class of the real functions defined on a h 1 -set contained in the real line, we prove the existence of functions whose graphs are h-sets, under appropriate assumptions for the mass function h. In particular, we obtain maximal box and Hausdorff dimensions of the graphs of Hölder functions on a d 1 -set contained in the 45

55 4.2 Dimensions and smoothness of some graphs 4 F. G. W. F. real line, and the maximal Hausdorff dimension over all real functions f B s p,q which are traces of continuous functions on a d 1 -set, for any 1 < p < and 0 < s < d Dimensions and smoothness of some graphs Consider ρ > 1, 0 < s 1 and let W s (x) := j 1 ρ js sin(ρ j x), 0 x 1, be a Weierstrass function, see Figure 12. Falconer gives (see [14], pp with elementary adaptations) the inequality dim B Γ(W s ) 2 s and for ρ ρ(s) large gives also dim B Γ(W s ) 2 s. More general Weierstrass-type functions occur as invariant sets in dynamical systems, see [14], p. 196, exercise 13.7, and it is our aim here to give results for them concerning box and Hausdorff dimensions. W s (x) Figure 12: Graph of the x Weierstrass function W s (x) = j 1 2 js sin(2 j x) with s = 0.6 Functions with an uniform fractal structure on their supports tend to be localized in a horizontal line in the ( 1 p, s)-plane, as does the function f s of Theorem As a consequence of Theorem 4.2.1, which is a trigonometric counterpart of Corollary 9 of [37], p. 202, the same occurs in the case of the function ψw, where ψ is a non-zero infinitely smooth cutoff function of the Schwartz class and W is a trigonometric Weierstrass-type function. Theorem Let 0 < p, 0 < q and s R. Consider (a j ) j N l 1, (θ j ) j N R and ψ S(R n ) \ {0}, and define W : R n R with W (x 1,..., x n ) := j 1 a j cos(2 j x 1 + θ j ). (Uniform pointwise convergence.) Then we have the equivalence ψw B s p,q(r n ) (2 js a j ) j N l q. 46

56 4.2 Dimensions and smoothness of some graphs 4 F. G. W. F. Remark (a) If we consider (a j ) j N C such that a j c r 2 jr for some r > 0, relaxing the hypothesis (a j ) j N l 1, then we loose the pointwise convergence in the series W of Theorem 4.2.1, but we keep the convergence in S (R n ) with the strong topology. (In fact, since the series as in the Theorem converges strongly in S (R n ) then, by the arguments used in part (i) of the Proof of Theorem 3.6.3, we can conclude that the more general series relaxing (a j ) j N l 1 also converges in S (R n ).) Furthermore, although we have no longer pointwise convergence for this more general series W, we may apply exactly the same Proof of Theorem for this case, in particular it comes out that the equivalence of the Theorem remains true. (b) We can see in the Proof that (at least) the implication = of Theorem is valid also for the spaces Fp,q(R s n ) (0 < p < ), including part (a) of the present Remark. In fact, Formula (8) of the Proof of the Theorem remains true by interchanging the order of integration and summation. (c) If we replace cos(2 j x 1 + θ j ) by e i(2j x 1 +θ j ) then, as we can see in the Proof, the full equivalence of Theorem holds, including part (a) of the present Remark, for both B-spaces and F -spaces (0 < p < ), and the proof is simpler. In particular, we don t need to use Lemma in the proof. (In fact, the Formula (8) of the Proof of Theorem remains true, and the proof is simpler, when we replace cos(2 j x 1 + θ j ) by e i(2j x 1 +θ j ).) (d) Both Lemma and Theorem are true for all ϕ as in Definition Moreover, with the same Proof we have a lower counterpart for the Theorem, i.e., the equivalence ψw B s p, (R n ) lim j 2 js a j < holds. The previous parts (a) and (c) of the present Remark also hold for these lower classes B s p, (R n ). (Cf. remark (c) of Definition and remark (c) of Definition ) The following Theorem gives estimates for box dimensions of trigonometric Weierstrasstype functions. Together with the equality of Theorem 4.4.3, this Theorem generalizes the result given in [14], pp Concerning box dimension of trigonometric 47

57 4.2 Dimensions and smoothness of some graphs 4 F. G. W. F. or more general Weierstrass-type functions, see for instance [1, 21]. Theorem Consider ρ > 1, (θ j ) j N R, s := (s j ) j N R + { }, s := lim j s j > 0 and s := lim j s j. Let W s : [0, 1] R be defined by W s (x) := j 1 ρ js j cos(ρ j x + θ j ), where ρ = 0. Then we have (a) dim B Γ(W s ) = 2 min{1, s}, (b) dim B Γ(W s ) 2 min{1, s}. Remark (a) Consider W s : R R defined as in Theorem and let ψ : R R with ψ S(R) \ {0}. Then, as it is quoted in the Proof of the Theorem, we have the relations ψw s Bp ρ,s (R) = Bp s (R) and ψw s B ρ,s p (R) for 0 < p, see Definition below. Moreover, by Theorem with elementary adaptations then the equalities dim B Γ(ψW s ) = dim B Γ(W s ) and dim B Γ(ψW s ) = dim B Γ(W s ) hold. (b) If 0 < s < 1 and W s (x) := j 1 ρ js cos(ρ j x + θ j ) then Theorem 4.2.3, part (a) of the present Remark, and remark (b) of Definition give the equality dim B Γ(ψW s ) = dim B Γ(W s ) = 2 s. By the previous part (a), it also holds ψw s Bp ρ,s (R) = Bp s (R) and ψw s B ρ,s p (R) for 0 < p. (c) In Theorem (b), the inequality does not hold. In fact, if 0 < s < 1 and W s (x) = j 1 4 js cos(4 j x), then part (b) of the present Remark (applied to ρ = 4) gives dim B Γ(ψW s ) = 2 s. Nevertheless we have s = for W s (x) = j 1 2 (2j)s cos(2 (2j) x), in other words ψw s B s p, (R) = B 2,s p, (R) for any s R, in particular for large positive s. Now we switch to the Hausdorff dimension, whose estimation is much more difficult in the general case, and particularly in what concerns the graphs of the Weierstrasstype functions. Let us start by recalling the following familiar Lemma, as well as the result presented in Theorem below, which concerns the Hausdorff dimension of the Weierstrass function in the statistical case. Proposition Consider ζ, f : [0, 1] R where f is bounded and ζ is a Lipschitz function, and let T : Γ(f) Γ(ζf) be defined by T (x, f(x)) := (x, (ζf)(x)). Then T 48

58 4.2 Dimensions and smoothness of some graphs 4 F. G. W. F. is a contraction, i.e. T (x, y) T (x, y ) c (x, y) (x, y ), (x, y), (x, y ) Γ(f). Proof. Consider x, x [0, 1]. Then, we have (ζf)(x) (ζf)(x ) = ζ(x)f(x) ζ(x )f(x ) ζ(x)f(x) ζ(x )f(x) + ζ(x )f(x) ζ(x )f(x ) c x x sup f + sup ζ f(x) f(x ) c ( x x + f(x) f(x ) ) c (x, f(x)) (x, f(x )). Remark: As T (Γ(f)) = Γ(ζf), then [14], p. 30, Corollary 2.4(a), gives dim H Γ(ζf) dim H Γ(f). (Cf. Theorem concerning box dimensions.) Theorem (Theorem 1 of [16], with elementary adaptations) Consider ρ > 1, 0 < s < 1, and let θ := (θ j ) j N [0, 2π] with an uniform measure on [0, 2π]. Let W s : [0, 1] R be defined by W s (x) := j 1 ρ js cos(ρ j x + θ j ). Hence, dim H Γ(W s ) = 2 s for almost all sequences θ, where the set of all sequences θ, := [0, 2π] N, is endowed with the measure induced by the Lebesgue (uniform) measure normalized by the factor 1, so the set has measure 2π 1. By almost all we mean all θ \ for some with measure zero. Proof. We summarize here the main ideas of the Proof given in [16], in order to highlight why the Theorem is not deterministic. Fix 0 < t < 2 s and set W θ := W s and 1 Φ θ (x, y) :=. First it is proved that Φ ((x y) 2 +(W θ (x) W θ (y)) 2 ) t/2 [0,1] [0,1] θ(x, y)dxdydθ <. This is made by changing the order of integration, estimating much more easily ( Φ [0,1] [0,1] θ(x, y)dθ ) dxdy <. Secondly, because ( ) Φ [0,1] [0,1] θ(x, y)dxdy dθ <, then we conclude that Φ [0,1] [0,1] θ(x, y)dxdy < for almost all sequences θ, so by the Theorem 4.13(a) of [14], pp , we obtain dim H Γ(W θ ) t and afterwards dim H Γ(W θ ) 2 s for almost all sequences θ. As it is well known that the converse of the last inequality holds for all sequences θ, then we obtain the desired result. By considering Remark (b) with some adaptations, we see that this value 2 s coincides with the box dimension of Γ(W s ). Furthermore, box and Hausdorff dimensions keep the same value for the graph of the product ψw s (see remark of Proposition with elementary adaptations, concerning Hausdorff dimension), where ψ is a non-zero 49

59 4.2 Dimensions and smoothness of some graphs 4 F. G. W. F. infinitely smooth and real cutoff function of the Schwartz class S(R). Moreover, for any integrability 0 < p, the exact smoothness of this function ψw s is just s for all sequences of phases. (We notice that for the function W s we have this smoothness s only for an integrability p =, whereas with the introduction of a cutoff function ψ we obtain, with the function ψw s, a much better relation between dimensions and smoothness.) Theorem remains valid if we replace the constant s by any sequence (s j ) j N R + such that s := lim j s j satisfying 0 < s < 1 cf. Theorems and and Remark (a). As we can see in Theorem 3.5.5, if 1 p and 0 < s < 1 then that value 2 s is maximal for the Hausdorff dimension on the class of all continuous real functions with exact smoothness s, so Theorem is important in what concerns showing the existence of functions that achieve that maximal value. This Theorem can be generalized also by replacing the cosine function by a more general real, Lipschitz and non-constant periodic function g satisfying some additional hypotheses, according to Theorem 2 of [16]. One inconvenient, though, is that it is not deterministic, so even for the typical case ρ = 2 and θ = 0, the Hausdorff dimension of Γ(W s ) for the so called Weierstrass function is still unknown. Despite that fact, in [23], p. 800, Theorem 8, it is given (deterministically) for the Hausdorff dimension of the graph of W s, that dim H Γ(W s ) 2 s c/ log ρ, for all sequences θ, where c does not depend on ρ, so the right hand side grows slowly to 2 s when ρ tends to infinity. (Recall, on the other hand, the well-known relation dim H Γ(W s ) dim B Γ(W s ) dim B Γ(W s ) 2 s.) Though the Proof of the inequality above uses techniques which are very different from the ones employed in [16], both references use similar measures, respectively induced by Cantor and Lebesgue measures. Several authors have discussed the complexity of the graphs of Weierstrass functions, showing how cumbersome is to make calculations which depend on the behavior of those graphs. So, it is not a surprise that the knowledge of the true value of dim H Γ(W s ) remains an open problem. As we will see in Remark below, by using similar but simpler techniques than 50

60 4.2 Dimensions and smoothness of some graphs 4 F. G. W. F. the ones employed in [23] we will repeat and extend the estimation of the lower bound 2 s c log ρ given in that reference. Moreover, by applying those techniques we prove the deterministic result stated in Theorem 4.2.7, which gives the Hausdorff dimension of scale-sparse series, coinciding with the lower box dimension. Theorem Let g : R R be a periodic Lipschitz function, satisfying (g(x) g(y)) x,y (x y) for x and y belonging to some subinterval of the real line, and (g(x) g(y)) x,y (x y) on another subinterval. Consider ρ > 1, γ > 1, 0 < s < 1 and (θ j ) j N R. Let W s : [0, 1] R be defined by W s (x) := j 1 ρ γjs g(ρ γj x + θ j ). Then, for any sequence (θ j ) j N, we have the equality dim H Γ(W s ) = 2 Ξ(γ, s), γs where Ξ(γ, s) := 1 s+γs. We observe that for fixed γ > 1, the one variable function Ξ(γ, ) is strictly increasing and ranges over (0, 1) for 0 < s < 1; analogously, for fixed 0 < s < 1, the function Ξ(, s) is strictly increasing and ranges over (s, 1) for 1 < γ <. Theorem is an extension of a result given in [2], where the case when g is a triangular wave was considered (think for instance in the distance to the nearest integer) and θ j = 0 for all j N. We must observe that by recalling Theorem 3.5.3, and by using remark (a) of Definition to compare dimensions, we can now easily conclude for the lower box dimension the same value dim B Γ(W s ) = 2 Ξ(γ, s) = 2 γs 1 s+γs = s. 1 s+γs Moreover, an easy estimation of the oscillations of W s, by using only standard arguments, shows that the equality dim B Γ(W s ) = 2 s holds. It is also convenient to notice that the Theorems and 4.2.7, as well as the comments and estimates following them, also hold for the limiting case s = 1; in other words, for this smoothness s = 1 we have dim H Γ(W s ) = dim B Γ(W s ) = 1. This follows from the equality dim B Γ(W s ) = 1 for this limiting case and the remark (a) of Definition As a special case of Theorem 4.2.7, it comes out that, for all κ > 0 and 0 < s < 1, the 51

61 4.2 Dimensions and smoothness of some graphs 4 F. G. W. F. function of [33], p. 121, when the graph is in R 2, f s (x) = j 1 2 ν js k Z ω(2ν j x k), where ν j := 2 jκ, actually has Hausdorff dimension 2 Ξ(2 κ, s). (See at the beginning and at the last part of Section 4.4, especially Figures 13 and 16.) Furthermore, the equalities above give the same value for the lower box dimension, so both are strictly smaller than the upper box dimension 2 s. In [25], p. 74, we find a result analogous to that one of Theorem 4.2.7, but it concerns wavelet series and is stated only in a probabilistic form; however, we believe that a (deterministic) counterpart for Theorem concerning wavelet series also holds. Remark (a) Let (λ j ) j N R + be such that γ := lim j λ j+1 λ j satisfying γ > 1, and consider (s j ) j N R + such that s := lim j s j satisfying 0 < s < 1. Let W s : [0, 1] R be defined by W s (x) := j 1 2 λ js j g(2 λ j x + θ j ), where (θ j ) j N R and g is as in Theorem Hence, the equality of that Theorem remains true for these γ and s and this more general scale-sparse Weierstrass-type function. (b) Consider ρ > 1, 0 < s < 1, (θ j ) j N R, and g as in Theorem Let W s : [0, 1] R be defined by W s (x) := j 1 ρ js g(ρ j x + θ j ). Then, we have dim H Γ(W s ) 2 s c g,s log ρ, where c g,s does not depend on ρ. This estimation is given by Mauldin and Williams in [23], Theorem 8, p. 800, but it can be proved also by using the same techniques applied in the Proof above, including the generalization when we replace the constant s by any sequence (s j ) j N R + such that s := lim j s j satisfying 0 < s < 1. (b ) Moreover, let W s := j 1 ρ s j j g(ρ j x + θ j ), with ρ := lim j ρ j+1 ρ j > 1, ρ := lim j ρ j+1 ρ j < and s := lim j s j satisfying 0 < s < 1. Then dim H Γ(W s ) 2 s c g,s / log ρ. This remains true even when ρ = as long as lim j λ j+1 λ j = 1, where 2 λ j dim H Γ(W s ) = 2 s. := ρ j ; in particular, if ρ = ρ = then dim H Γ(W s ) 2 s 0 and so (Application: Taking W s (x) := j 1 ρ s k k =1 a k j r k g(ρ k k =1 a k j r k x + θ j ), with ρ > 1, 0 < s < 1, a 1,..., a k > 0 and r 1,..., r k > 1, then dim H Γ(W s ) = 2 s holds.) (c) Consider ρ > 1, 0 < s < 1, (θ j ) j N R, and Λ a triangular wave. Let 52

62 4.3 Comparing box and Hausdorff dimensions 4 F. G. W. F. W s : [0, 1] R be defined by W s (x) := j 1 ρ js Λ(ρ j x + θ j ). If ρ ρ(s, ε) is large (in particular satisfying ρ 1 s > 2), then the inequality dim H Γ(W s ) 2 s cρ,s,ε log ρ holds with c ρ,s,ε := 1 s s 1 ρ 1 s 2 + ε, whenever ε > Comparing box and Hausdorff dimensions In remark of Definition were given some basic properties concerning box and Hausdorff dimensions. The following Proposition allow us to get a more precise idea about these dimensions in terms of monotonicity and stability. We intend to compare box and Hausdorff dimensions when applied to graphs of functions, since we are now in a position to do it by considering the knowledge we acquired in the previous Sections. Proposition (see [14]) (a)(open sets and smooth sets) If F R n is open then it is a n-set, so dim H F = dim B F = n. On the other hand, if F is a smooth (i.e. continuously differentiable) m-dimensional surface of R n, then F is an m-set. (b)(monotonicity) If F 1 F 2 R n then dim B F 1 dim B F 2, and analogous inequalities hold for lower box and Hausdorff dimensions. (c)(finite stability) Suppose that F 1, F 2 R n. Then we have dim B (F 1 F 2 ) = max{dim B F 1, dim B F 2 }. Nevertheless, the corresponding counterpart for the lower box dimension does not hold. (d)(countable stability) If (F j ) j N R n then dim H Fj = sup dim H F j. On the other hand we have no counterpart for box dimensions; for instance we have dim B {q} = 0, q Q, and nevertheless dim B Q = 1. (e)(translations) If a R n and F R n, then of course we have dim B F = dim B (a+f ), where a + F := {a + y : y F }, as well as the analogous identities for lower box and Hausdorff dimensions. As we already saw at the end of Section 1.1, there are continuous functions whose graphs establish a striking distinction between box and Hausdorff dimension. Next, we 53

63 4.3 Comparing box and Hausdorff dimensions 4 F. G. W. F. will emphasize this distinction, with the smoothness coming also into play. Let 0 < h n. Then there is a continuous real function Θ h, which does not depend on p, satisfying Θ h B h p p (R n ) for 0 < p, such that dim B Γ(Θ h ) = n + 1. This follows by Theorem applied to Θ 0 as in Definition However, it is easy to check that the equality dim H Γ(Θ h ) = n holds. In fact, set E 0 := {0} R n 1 by technical convenience we may assume n 2, E g := (R \ ] 1/g, 1/g[ ) R n 1 if g N, and define Γ g := Γ(Θ h Eg ) for all g N 0. Of course we have dim H Γ 0 = n, because Γ 0 {0} R n 1 R R n. On the other hand, for each g N the restriction Θ h Eg is a function with a continuous derivative uniform convergence of the derivative series, see Definition 3.4.2, so we have dim H Γ g = n. This follows by remark (c) of Definition 2.2.4, Theorem (a) and second part of remark (a) of Definition 2.5.3, with elementary adaptations. As we have Γ(Θ h ) = g N 0 Γ g, then by Proposition (d) the Hausdorff dimension of the graph Γ(Θ h ) is n. But there is a bit more to quote. We state that there is a real, continuous, and compactly supported function Ω h, which does not depend on p, satisfying Ω h B h p p (R n ) for 0 < p and 0 < h n, for which we have dim B Γ(Ω h B ) = n + 1 for every ball B R n centered at any P [0, 1] n, and nevertheless dim H Γ(Ω h ) = n holds as soon as h 1. This follows by Theorem below, applied to Ω h of the following Definition Analogously to the mentioned above concerning Definitions 3.4.1, and 3.4.3, the representation in wavelets is here a reasonable alternative for the construction of Definition Definition Let ψ : R n R with ψ S(R n ) \ {0} and supp(f ψ) {ξ R n : 1 a ξ 1 + a}, for a > 0 sufficiently small. Consider 0 < h n. For each j N let d(j) := [2 h = n), and for l {1,..., d(j) n } let m(j, l) := n h j2 n ( s1,l d(j),..., s n,l d(j) ] (modification d(j) := [2 j ] if ), where the d(j) n different values of l correspond to the different values of (s 1,l,..., s n,l ) {1,..., d(j)} n. ( ) Let Ω j,l : R R be defined by Ω j,l := 1 ψ 2 j2 ( m(j, l)). j 2 54

64 4.3 Comparing box and Hausdorff dimensions 4 F. G. W. F. Let Ω : R n R with Ω := d(j) n j 1 l=1 Ω j,l. (Uniform convergence.) Let φ : R n R with φ S(R n ) and satisfying φ(x) = 1, x [0, 1] n. Let Ω 0 : R n R with Ω 0 := φ Ω. Theorem Let 0 < h n and Ω 0 as in Definition Then (a) Ω 0 is bounded and continuous, and Ω 0 B h p p (R n ) holds for 0 < p. (b) dim B Γ(Ω 0 I ) = n + 1 for all I [0, 1] n as in Definition oscillation over the cubes I satisfies osc I (Ω 0 I ) c I s I with lim I 0 s I = 0. (c) dim H Γ(Ω 0 ) n + 1 h if 0 < h 1, and dim H Γ(Ω 0 ) = n if 1 h n. Moreover, the Proof. The proofs of parts (a) and (b) are analogous to that corresponding ones for Theorem 3.4.4, with some adaptations. On the other hand, for p = h the part (a) gives the relation Ω 0 B 1 h (Rn ). Then, by applying part ( ) of Theorem given with reference to the Theorem 4.8 of [25], p. 77 we obtain dim H Γ(Ω 0 ) n + 1 min{1, h}. What shows a striking distinction between lower and upper box dimension are the graphs of scale-sparse series. Namely, concerning the function W s as in Corollary 3.5.4, its graph has lower box dimension n, but an estimation of the oscillations of this function, by using standard arguments, shows that it has upper box dimension n + 1 s, which is remarkable since this value tends to n + 1 when s 0 +. More generally, big gaps in the frequency domain lead to lower box (and Hausdorff) dimension strictly smaller than the upper box dimension. On the other hand, big gaps in the space domain, as occours in the graph of Ω 0 above, lead to a Hausdorff dimension strictly smaller than box dimensions. Naturally, box and Hausdorff dimensions behave distinctly, since the later is based in measures and the formers depend on the oscillations of the function instead. Because Weierstrass functions do not have gaps neither in the domain nor in the frequence, box and Hausdorff dimensions typically coincide for them, while in the scale-sparse case the lower box and Hausdorff dimensions coincide and both could be strictly smaller than 55

65 4.4 Search and construction of d-sets 4 F. G. W. F. the upper box dimension. Box dimensions are very useful and widely used in practice and theory, mainly due to the fact they are easy to calculate and intimately related with smoothness in the oscillation spaces. For a first approach, it is very convenient to start with box dimensions, since they allow us to make estimates for the much more cumbersome Hausdorff dimension, which are sharp in many interesting cases. On the other hand, Hausdorff dimension satisfies the countable stability property, which is very convenient for theoretical purposes, while box dimension does not satisfy this property (and not even the finite stability property in the lower case). 4.4 Search and construction of d-sets This Section mainly deals with geometrical concepts, so not surprisingly we employ many geometrical arguments. Calculations for Hausdorff dimension carry with them the difficult task of dealing with the Hausdorff measures. However, see remark of Definition 2.5.5, especially part (a), these can be handled easily when Γ(f) is a d-set, since dim H Γ(f) = dim B Γ(f) = d holds in this case. Actually, these graphs are sets with special properties that can be advantageous in many circumstances, so the goal of the present Section is to construct functions on [0, 1] n the graphs of which are d-sets, for each d between n and n + 1. Graphs of Weierstrass-type functions present a good uniformity of fractal structure, thus they can play an important role when we are interested in d-sets or Hausdorff dimension. (We will discuss the importance of this kind of graphs at the beginning of the next Section 4.5, where we refer some of their most remarkable properties.) Remark: Let ζ : R R be a bounded and Lipschitz function. Consider 0 < s 1, κ > 0 and ν j := 2 jκ, j N. Let f s : [0, 1] R with f s (x) := j 1 2 ν js ζ(2 ν j x), see Figure 13. Then by Theorem we have dim B Γ(f s ) 2 Ξ(2 κ, s) = 2 2κ s 1 s+2 κ s. If κ > 0 and 0 < s < 1 then by the last inequality and by remark (a) of Definition we have also 1 dim H Γ(f s ) dim B Γ(f s ) 2 Ξ(2 κ, s) < 2 s. 56

66 4.4 Search and construction of d-sets 4 F. G. W. F. In particular, since dim B Γ(f s ) = 2 s then by remark (a) of Definition it follows that Γ(f s ) cannot be a d-set for any d. Actually, these inequalities by Theorem and comments following it, the second and third are actually equalities give a contradiction with the following claim, that comes out from [33], pp , specially p. 121, when the graph is in R 2 : CLAIM: Let ζ 0 C (R) be defined by ζ 0 (x) := 1 k Z ω(x k), with ω(x) := e 1 4x 2 x < 1 2 and ω(x) = 0 otherwise. Consider 0 < s < 1, κ N with 2κ s > 1, ν j := 2 jκ and f s : [0, 1] R with f s (x) := j 1 2 ν js ζ 0 (2 ν j x). Then Γ(f s ) is a d-set where d = 2 s, and consequently by remark (a) of Definition we have dim H Γ(f s ) = 2 s. if In the following, we show where this contradiction has originated, concerning the function f s, see Figure 13. In order to do this, we start by stating that the second relation of (16.21) in p. 122 of [33] cannot be true, i.e. µ(b r ) r r d where d = 2 s and B r is a ball with B r = r and centered at any P Γ(f s ) = supp µ, cannot be true for all 0 < r < 1. Moreover, the procedure of the uniform mass distribution described in [33], p. 122, is inconsistent. Furthermore, the relation µ(b r ) r r d, with d = 2 s and 0 < r < 1, cannot be true, for any measure µ with Γ(f s ) = supp µ. And, as we will see below in Theorem (which concerns a much wider class of Weierstrass-type functions), that relation is also impossible for any 1 d 2, therefore Γ(f s ) cannot be a d-set for any d (as we already concluded in comments that follow the inequalities above). 57

67 4.4 Search and construction of d-sets 4 F. G. W. F. f s (x) B j : diameter c2 ν js Ball B j : diameter j 2 ν j Ball B j: diameter j 2 ν j(s+2 κ (1 s)) Angle α j 1 : tan α j 1 j 2 ν j 1(1 s) α j 1 Figure 13: Graph of f s (x) = j 1 2 ν js ζ 0 (2 ν j x), where 0 < s < 1, ν j = 2 jκ with κ N such that 2 κ s > 1, and ζ 0 C 1 (R) is defined by ζ 0 (x) = 1 k Z ω(x k), with ω(x) = e 1 4x 2 x if x < 1 2 and ω(x) = 0 otherwise Before state and prove Theorem we point out the following calculations and geometrical arguments, which can be easily generalized. Taking any ball B j according to Figure 13, with B j j 2 ν j, suppose that µ(b j ) j B j d j 2 νj(2 s). Consider now appropriate balls B j, looking at Figure 13 again, as well as the angle α j 1 in the same 58

68 4.4 Search and construction of d-sets 4 F. G. W. F. picture. Then tan(α j 1 ) j sup ( j 1 ) j =1 2 ν j s ζ 0 (2 ν j ) j 2 ν j 1(1 s) holds, where ( ) represents the classic derivative operator. Therefore, we can write B j j 2 ν js cos(α j 1 ) j 2 ν js 1 tan(α j 1 ) j 2 ν js 2 ν j2 κ (1 s) j 2 ν jσ, where σ := s + 2 κ (1 s) satisfies s < σ < 1. Hence, since µ is a measure and looking to Figure 13, taking into account the particular characteristics of the graph, we obtain ) 2 µ(b j) j µ(b j ) j B j 2 2 νj(2 s) 2 2ν j j B j 2 (2 νjσ ) s σ j B j 2 s σ. ( B j B j So, with Ξ(2 κ, s) = s2κ 1 s+s2 κ = s σ given as in Theorems 3.5.3/4.2.7 it holds µ(b j) j B j 2 Ξ(2κ,s). Hence this relation contradicts the claim µ(b r ) r r d with d = 2 s in (16.21) of [33], p Moreover, by slightly more general calculations and the same geometrical arguments, on the basis of Figure 13, we have also the following Theorem. Theorem Let ζ : R R be a non-constant, periodic and Lipschitz function. Consider 0 < s < 1, let (ρ j ) j N R + be such that ρ j+1 ρ j c ζ,s sufficiently large, and assume that the sequence ( ρj+1 ρ j )j N W s : [0, 1] R defined by W s (x) := j 1 ρ s j ζ(ρ j x + θ j ). Then is unbounded. Consider (θ j ) j N R and Γ(W s ) is not a d-set for any 1 d 2. Proof. In fact, if Γ(W s ) were a d-set then by Theorem and remark (a) of Definition we would have d 2 s, so d < 2. On the other hand, by using the arguments above, based in Figure 13, but with B j j ρ 1 j, tan(α j 1 ) j ρ 1 s j 1 and B j j ρ s j /ρ 1 s j 1 = ( ρ 1 j (ρ j /ρ j 1 ) 1 s instead, then it would hold µ(b j)/µ(b j ) j B j / B j ) 2, besides the relation µ(b j)/µ(b j ) j ( B j / B j ) d. But this is impossible, because the fraction B j / B j j (ρ j /ρ j 1 ) 1 s is unbounded. As a matter of fact, the graph Γ(W s ), which is a subset of R 2, does not even satisfy the ball condition, i.e. for any η (0, 1), there exists a ball B rη R 2 (with diameter B rη = r η ) such that B ηrη Γ(W s ) for all B ηrη B rη (with diameter B ηrη = ηr η ), 59

69 4.4 Search and construction of d-sets 4 F. G. W. F. denoting this relation by (NBC). In fact, for any η (0, 1), this relation holds by taking an appropriate ball B rη := B j η as in the Proof above, where j η N is chosen such that ( ) 1 s ρ jη 1/ρ jη c0 η for some fixed and sufficiently small c 0 > 0, so the balls B ηrη can be written as B ηrη = c η B jη for some c η 1, in order to satisfy the equivalence ( ) 1 s c η ρjη 1/ρ jη η η and, more precisely, the equality B ηrη / B rη = η. Proposition The relation (NBC) above, concerning η (0, 1) and B rη and B ηrη, implies that Γ(W s ) cannot be a d-set for any d < 2. More generally, if a given set E R 2 does not satisfy the ball condition, then E cannot be a d-set for any d < 2. (Equivalently, if E is a d-set and does not satisfy the ball condition, then d = 2.) Proof. If E does not satisfy the ball condition, according to the relation (NBC) above with E in place of Γ(W s ), then there is a ball B rη centered at a point of E with the following property: there are η (1/η) 2 disjoint balls B ηrη contained in B rη, with diameter ηr η and centered at points of E; therefore it holds µ(b ηrη )/µ(b rη ) η η 2. On the other hand, if E is a d-set then we have µ(b ηrη )/µ(b rη ) η η d ; consequently we obtain η d η η 2 for all η (0, 1), and this means d = 2. Of course, this easy and short Proof is immediatly extendable to subsets of R n ; moreover, we may equivalently replace balls by cubes, in particular the Proof would come simpler. See also Proposition 4.3 of [7], pp , concerning when the result in Proposition above was noticed firstly. (Cf. [6], p. 54 and [34], , pp ) Concerning the function f s as in Figure 13, we will give at the end of the present Section additional details for the behavior of a mass distribution µ under the assumption d = 2 s and µ(b j ) j B j d j 2 νj(2 s) for B j j 2 ν j. Now, we return to the main point. Remark: We proved that the claim above, concerning the function f s, is false. Nevertheless, in the present Section we will state that Theorem 16.2 and Remark 16.3 of [33], pp. 120 and 122, both are true. We only need a better function, instead of that one f s, for the proofs. Functions whose graphs are d-sets, with n < d < n + 1, do not occur frequently. More- 60

70 4.4 Search and construction of d-sets 4 F. G. W. F. over, as we said at the beginning of the Section, such functions satisfy dim H Γ(f) = d, however (cf. [23], pp ) the graphs of almost all real functions f on R n have Hausdorff dimension n. So, we must search for them carefully in the class of good candidates. And, as we can see in Corollary below, even promising candidates may not have the desired property. Theorem Consider ρ > 1, 0 < s < 1, (θ j ) j N R, ζ : R R a nonconstant, periodic and Lipschitz function and W s : [0, 1] R defined by W s (x) := j 1 ρ js ζ(ρ j x + θ j ). Let µ 0 (U) := λ ({x [0, 1] : (x, W s (x)) U}) for Borel sets U R 2, where λ is the Lebesgue measure, and suppose that B r are balls with 0 < B r = r < 1 and centered at any P Γ(W s ). If ρ ρ(ζ, s) is large then we have the equality dim B Γ(W s ) = 2 s and the implication Γ(W s ) is a d-set = [d = 2 s and µ 0 (B r ) r r 2 s ]. Definition A Lipschitz function Λ : R R, with period 1, is a triangulartype wave, if the quantity (Λ(x) Λ(y))/(x y), where x y, is positive for x, y [0, 1/2], negative for x, y [1/2, 1], and has in both cases an absolute value greater than c > 0 for x, y in a neighborhood of 1/2. 61

71 4.4 Search and construction of d-sets 4 F. G. W. F. W s (x) Ball B j, with diameter j ϱ j x Figure 14: Graph of W s (x) = j 1 ϱ js Λ(ϱ j x), with ϱ = 2k + 1 according to Corollary with Definition We show the case ϱ = 9 and s = 0.6, with a triangular wave Λ Corollary Consider ϱ := 2k + 1 with k N, 0 < s < 1, let Λ be a triangulartype function according to Definition 4.4.1, and let W s W s (x) := j 1 ϱ js Λ(ϱ j x), see Figure 14. If ϱ ϱ(s) is large then Γ(W s ) is not a d-set for any 1 d 2. : [0, 1] R be defined by Proof. Consider j N and let B j be a sufficiently small ball with diameter B j j ϱ j ( kϱ and centered at a point j, W 2 s( ), kϱ j ) k {0,..., 2ϱ j } odd, of local maximum of 2 W s, see Figure 14. Then, there is an unique peak contained in the ball B j, with a height of j ϱ m js for some m j N satisfying ϱ m js j ϱ j. Therefore, the width of the peak is j ϱ m j = (ϱ m js ) 1 s j (ϱ j ) 1 s. Let now µ 0 (U) := λ ({x [0, 1] : (x, W s (x)) U}) for Borel sets U R 2. Then, by definition of µ 0 we have µ 0 (B j ) j width of the peak j (ϱ j ) 1 s j B j 1 s. But we have 1 s > 2 s, therefore by Theorem the graph Γ(W s) cannot be a d-set. Remark With similar proofs, we can extend Theorem and Corollary even for h-sets. 62

72 4.4 Search and construction of d-sets 4 F. G. W. F. (a) Let W s be given as in the Theorem. If Γ(W s ) is a h-set then it holds the relation µ 0 (B r ) r h(r). This follows by an appropriate refinement of the Proof of the Theorem. (b) Let W s be given as in the Corollary. Hence, the graph Γ(W s ) is not a h-set, for any h according to Definition In fact, if that graph was a h-set then by part (a) and by the Proof above we would have h(r) r µ 0 (B r ) r r 1 s, what is impossible because dim B Γ(W s ) = 2 s < 1. s Definition Let V N \ {1}, H 0 := 2k with k 0 N, H := H 0 V and the ( ) sequence W (j) V,H of functions W (j) V,H : R R be defined as follows. j N Consider 0 x 2 and start with W (0) V,H according to Figure 15a, focusing the attention in the square [0, 1] 2 that just covers the segment given by the graph of W (0) V,H [0,1]. First iteration: We divide that square, both vertically and horizontally in V N \ {1} equal parts, originating V 2 identical rectangles though only V of them contain segments as diagonal lines, see Figure 15a for V = 4. After that we divide vertically each of these V rectangles into identical H 0 = 2k rectangles, see Figure 15b for V = 4 and H 0 = 3. Next we transform each of those V diagonal lines into the union of H 0 connected diagonal lines, according to Figure 15b. We obtain W (1) V,H [0,1], see Figure 15b, having that the projection on the (horizontal) X-axis of the square that contains the original segment of W (0) V,H [0,1] was divided into H = H 0 V equal parts and the corresponding projection on the (vertical) Y -axis into V equal parts. We proceed similarly on [1, 2] and obtain W (1) V,H the Figure 15a originates Figure 15b if we have V = 4, H 0 = 3, so H = 12. This completes the first iteration giving W (1) V,H, see Figure 15b. Second iteration: We make the second iteration by repeating the process applied to the segment W (0) V,H [0,1], as described above, to each one of the H segments that constitute the graph W (1) V,H [0,1] which have equal lengths, and by proceeding similarly on [1, 2]. Hence we obtain W (2) V,H, whose graph on [0, 1] can be decomposed on H2 segments of equal lengths, and similarly on [1, 2], see Figure 15c. 63

73 4.4 Search and construction of d-sets 4 F. G. W. F. Iterations and limit: And so on, in the j th iteration we repeat the process to each one of the H j 1 segments of W (j 1) V,H [0,1], obtaining W (j) V,H [0,1], whose graph can be decomposed on H j segments, and similarly on [1, 2]. Then the j th iteration gives W (j) V,H, furthermore we extend each W (j) V,H by 0 to all R. Therefore we arrive to a function W V,H : R R defined by (uniform convergence) W V,H := lim j W (j) V,H. Remark: We notice that on [0, 1] the difference W (1) V,H W (0) V,H = V l=1 ±A(1) l can be written as a sum of V non-smooth atoms ±A (1) l, according to Definition 2 of [36], p. 465, where each A (1) l is given by a translation A (1) l := A( l 1 ) and A is a continuous non-smooth V atom with A(0) = 0 and A( 1 V ) = 0, which can be written as A(t) := (H 0 1)t if 0 t 1 H, A(t) := c 1 (H 0 + 1)t if 1 H t 2 H, A(t) := c 2 + (H 0 1)t if 2 H t 3 H,..., A(t) := c H0 1 + (H 0 1)t if H 0 1 t H 0 H H = 1 V More generally, with j N, on [0, 1] the difference W (j+1) V,H, and A(t) := 0 otherwise. W (j) V,H = V H j l=1 ±A(j+1) l can be written as a sum of V H j non-smooth atoms ±A (j+1) l given by translations as A (j+1) l := V j A ( H ( )) j l 1 V, and on the interval [1, 2] the situation is similar. 1 W (0) V,H (x) x Figure 15a: According to Definition we start with W (0) V,H. We choose V = 4 64

74 4.4 Search and construction of d-sets 4 F. G. W. F. 1 W (1) V,H (x) x Figure 15b: First iteration W (1) V,H, according to Definition Here V = 4 and H 0 = 3 W (2) V,H (x) Figure 15c: Second iteration W (2) V,H, according to Definition 4.4.2, with V = 4, H 0 = 3 x In fact, Definition is equivalent to the definition given in (4.42) of [33], p. 22, according to the construction suggested by Figure 4.7 of this reference, case 0 < a 1 < 2. Likewise, the following Theorem is equivalent to Corollary 4.23 of [33], p. 24. However, here we use a different approach with arguments based on the mass distribution µ 0 as in Theorem Theorem Let V N \ {1}, H 0 = 2k with k 0 N, H = H 0 V and W V,H 1 according to Definition Consider s V,H := log H V =. Then the relation 1+ log H 0 log V W V,H C s V,H (R) = B s V,H, (R) holds and 65

75 4.4 Search and construction of d-sets 4 F. G. W. F. Γ(W V,H [0,1] ) is a d-set with d = 2 s V,H. Corollary Let 0 < s < 1. Then, there exists a real function W s C s (R n ) = B s, (R n ) such that Γ(W s [0,1] n) is a d-set with d = n + 1 s. Remark: We can confirm now that Theorem 16.2 and Remark 16.3 of [33], pp. 120 and 122, both are true. In fact, Remark 16.3 of [33] is true for the function W s, 0 < s < 1, according to Corollary Hence, by remark (a) of Definition and because W s C s (R n ), we have also dim H Γ(W s ) = n + 1 s, therefore Theorem 16.2 of [33] follows. Moreover, with 0 < s < 1 and W s B s, (R n ) as in Corollary 4.4.7, we can obtain the relation W s B s p, (R n ) for 0 < p. This follows by an argument similar to that one used in parts (i) of the Proofs of Theorems 3.3.4/6.3.7, with elementary modifications in order to apply the Hölder s inequality with in place of 1. On the other hand, by the equality dim H Γ(W s ) = n + 1 s and by using the cell ( ) of Theorem given with reference to Theorem 4.8 of [25], p. 77, we obtain also W s B s p (R n ) at least for 1 p. Concerning d-sets and boundaries, the construction of the graph of W s [0,1] n as in Corollary see the respective Proof in Section 5.6 can be applied to modified situations (16.4 of [33], p. 123). For example, the n-dimensional unit cube can be replaced by the n-dimensional unit sphere S n in R n+1 ; instead of the x n+1 -direction we may use the direction of the outer normal. There is a counterpart of the function W s [0,1] n now with S n instead of [0, 1] n, the resulting graph being a boundary Ω of a star-like domain Ω in R n+1. Estimations from below, concerning Hausdorff dimension of the graphs of functions, are still a difficult problem. However, in [2, 15, 16, 23] and others we find estimates concerning box and Hausdorff dimensions of Weierstrass-type functions, and in Theorem 66

76 4.4 Search and construction of d-sets 4 F. G. W. F and Remark above we gave deterministic results that extend some of those estimations for Hausdorff dimension. François Roueff in [25] developed many further results both in analytical and statistical cases, in particular he gives estimations for box and Hausdorff dimensions for the graphs of wavelet series. Additional explanation concerning f s of Figure 13: As we saw in Theorem and comments following the respective Proof, applied here to the particular case of f s given as in [33], p. 121, the graph Γ(f s ) is not a d-set for any d, and indeed this graph does not even satisfy the ball condition. We will try to give a clearer explanation concerning a mass distribution µ on the graph of f s, under the assumption µ(b j ) j B j 2 s j 2 ν j(2 s) for any ball B j with B j = r j j 2 ν j. As in calculations before Theorem 4.4.1, we would get the relation µ(b j) j B j 2 Ξ(2κ,s) = r 2 Ξ(2κ,s) j for appropriate balls B j with B j = r j j 2 ν jσ. (Recall that σ satisfies s < σ < 1.) As a consequence of this estimation of the mass µ(b j), under the assumption above, we would obtain µ(b j ) c B j 2 Ξ(2κ,s) for the balls B j, see Figure 13. Then, by considering for Γ(f s ) an efficient covering of balls B j and B j, we would get H 2 Ξ(2κ,s) (Γ(f 2 ν j s s )) B B j j 2 Ξ(2κ,s) + B B j j 2 Ξ(2κ,s) c B j µ(b j) + c B j µ(b j ) j µ(γ(f s )), since with no much overlap the last relation j would hold. We would have H 2 Ξ(2κ,s) (Γ(f s )) cµ(γ(f s )) <, so not surprisingly we would find again the estimation dim H Γ(f s ) 2 Ξ(2 κ, s), see at the beginning of the present Section. As a matter of fact, by Theorem the equality dim H Γ(f s ) = 2 Ξ(2 κ, s) actually holds. Furthermore, by looking to the respective Proof, as well to the Figure 17, we can see that there is a Cantor-like set K R with Hausdorff dimension 1, and a mass distribution µ 0 ( ) supported on Γ(f s K ) with the following property (here, B r represents any ball with 0 < B r = r < 1 which may be centered at any P Γ(f s K ) = supp µ 0 ): The mass µ 0 (B r ) can be asymptotically sharply estimated from above by a curve behaving like described in Figure 16. More precisely, we have actually the sharp es- 67

77 4.4 Search and construction of d-sets 4 F. G. W. F. timations µ 0 (B j ) c B j 2 s ε j and µ 0 (B j) c B j 2 Ξ(2κ,s) ε j for all balls Bj and B j centered at any point P Γ(f s K ), where ε(j) and ε (j) are strictly positive and satisfy lim j ε j = lim j ε j = 0. In fact, the restriction of f s to K means that we remove small intervals around the stationary points of the partial sum of the series of f s, where an overcharge of mass would be originated, thus balls like B j as in Figure 13 are not taken into account. It turns out that the beautiful Figure 16 describes asymptotically a sharp estimation, from above, for the growth of that appropriate mass distribution µ 0, supported on the graph of f s K. Furthermore, although we believe that Γ(f s K ) is not even a h-set, it illustrates an interesting and representative class of graphs. Number a(r): 2 Ξ(2 κ,s) a(r) 2 s r r 2 Ξ(2κ,s) µ(b r ) r r a(r) r r 2 s r j+1 r j r j r j 1 r Figure 16: Asymptotical qualitative growth of µ 0 (B r ) on Γ(f s K ), where 0 < B r = r < 1 and 0 < s < 1. Points of minimal mass occur for r = r j j 2 ν j, with µ 0 (B j ) j r 2 s j, and points of maximal mass occur for r = r j j 2 ν jσ, with µ 0 (B j ) j r 2 Ξ(2κ,s) j = r 2 s σ j 68

78 4.5 Existence and construction of h-sets 4 F. G. W. F. 4.5 Existence and construction of h-sets As we have seen in the Theorem and comments following it, there is in general some uncertainty in the relation between smoothness and dimensions. On the opposite side, a remarkable example where this relation is described deterministically by a simple formula is given in Theorem 4.4.6, Corollary and comments following it. Here, we constructed a family of functions whose graphs are d-sets, where d = 2 s so box and Hausdorff dimensions equal 2 s and s (0, 1) is the smoothness of the function at least for an integrability 1 p. In the present Section we will deal with a generalization of those graphs, which actually are specialized cases of the wide class of the Weierstrass-type functions. Let 1 p, 0 < q, and n < d < n+1, s := n+1 d, thus s (0, 1). Hence, by Theorem it holds the sharp estimation (also for box dimensions if s > n) p dim H Γ(f) d, for all continuous real functions f Bp,q(R s n ). In particular this holds for all continuous and real f C s (R n ) = B, (R s n ), which are bounded functions on R n and satisfy f(y) f(x) c s, where := y x. In fact that estimation appears first in [25, 26], where we can find a wide collection of results concerning fractal dimensions of graphs, including box and Hausdorff dimensions, as well as many other references. Those graphs which are d-sets satisfy many remarkable properties, in particular they fulfill the equality dim H Γ(f) = dim B Γ(f) = d, therefore they behave as extremal cases in the estimation above. They are special sets, because they are graphs of functions; they satisfy the properties of d-sets, so have an uniform distribution of mass, up to a strictly positive constant; and they show a good relation dimensions-smoothness, which can be described by the simple formula d = 2 s. By the way, we find pertinent applications using d-sets which are graphs of functions by different authors, see e.g. [36], pp , or [6], p. 61. However, as we saw in Section 4.4, the construction of this kind of graphs is not a trivial problem. Furthermore, in Theorem and Corollary we showed explicitly 69

79 4.5 Existence and construction of h-sets 4 F. G. W. F. that there also families of Weierstrass-type functions whose graphs are not d-sets or not even h-sets. Natural generalization directions comprehend, under appropriate smoothness conditions, graphs of continuous functions on R n which are h-sets, as well as h-sets belonging to spaces of functions the domain of which is a h 1 -set. Since we believe that they still behave as extremal cases, they allow us to conjecture, for these more general situations, sharp estimations for the dimensions of the graphs of continuous functions. More precisely, consider 1 < p <, 0 < d 1 1 and 0 < s < d 1, and consider K R as an arbitrary d 1 -set. (Notice that when we intend to characterize dimensions of real functions on the real line, we can assume without any loss of generality that they are supported on [0, 1], which in fact is an 1-set.) By taking into account the Corollary below, we conjecture that the maximal value of the Hausdorff dimension over all real functions f belonging to Bp,q(K) s and which are traces of continuous functions, is given by sup f dim H Γ(f) = d s, where the supremum would be actually a maximum. In other words, we conjecture that sup f B s p,q (K) dim H Γ(f) = d holds, where d is chosen such that 1 < d < d and the smoothness s is determined by s := d d. (Concerning the definition of the Besov spaces Bp,q(K), s where s > 0, 1 < p <, 0 < q, we refer e.g. [34], , pp ) Actually, we will prove this conjecture in Corollary below, which indeed generalize the cell ( ) of the Theorem above, case n = 1 and 1 < p <. Construction of h-sets as functions on R n : Let H be the class of all continuous monotone functions h : (0, 1] R + satisfying h(0 + ) = 0. Bricchi in [4] and [5] gives necessary and sufficient conditions for the existence of h-sets in the general case: There exists a h-set in R n+1 if and only if h(r) r h(r) for some h satisfying δ n+1 h(δr)/ h(r) for all 0 < δ, r 1. Moreover, this condition holds if and only if c 1 δ n+1 h(δr)/h(r) for all 0 < δ, r 1 (for some c 1 > 0). (In order to prove the later equivalence put l(r) := h(r)/r n+1, so it is sufficient to show 70

80 4.5 Existence and construction of h-sets 4 F. G. W. F. that if l(s) c l(r) for all 0 < s r 1, then l(r) r l(r) for some decreasing l on (0, 1]. But this holds with l(r) := sup s [r,1] l(s).) Here, we are not interested in the general case, but in those h-sets which are graphs of functions. More precisely we intend, under appropriate smoothness conditions, to provide (only) sufficient conditions for the existence of those h-sets which are graphs of continuous real functions on R n. (The necessary conditions are left as an open question.) Theorem (Sufficient condition) Let h H (and we may relax the monotonicity condition) be a function satisfying c 1 δ n+1 ε h(δr)/h(r) c 2 δ n+ε for all 0 < δ, r 1, for some ε > 0. Then, there is a function W h : R n R, satisfying W h (y) W h (x) c n+1 /h( ) for all x, y R n with 0 < 1 (as above, we consider := y x ), such that the graph Γ(W h [0,1] n) is a h-set. Remark: (a) The usual (d, Ψ)-sets satisfy the hypotheses of the Theorem for h(r) := r d Ψ(r). (More precisely, consider n < d < n+1, and Ψ : (0, 1] R + continuous and satisfying Ψ(r 2 ) r Ψ(r). Hence, c ε δ ε Ψ(δr)/Ψ(r) c εδ ε for all 0 < δ, r 1, for any ε > 0.) (b) In particular, if n < d < n + 1, there is a function W d : R n R satisfying W d (y) W d (x) c n+1 d, such that Γ(W d [0,1] n) is a d-set. (Cf. Corollary and Remark following it.) (c) Case n = 1: If c 1 δ 2 ε h(δr)/h(r) c 2 δ 1+ε, then W h : R R, satisfying W h (y) W h (x) c 2 /h( ), such that Γ(W h [0,1] ) is a h-set. This case will be generalized in Theorem below. Construction of h-sets as functions on a h 1 -set R: Next, we will extend Theorem by giving sufficient conditions which guarantee the existence of h-sets belonging to spaces of functions the domain of which is a h 1 -set. The construction of such graphs are now achieved by applying the domain operator see Definition and Figure 18b in Section 5.7 below to the graphs obtained above on R, according to remark (c) of Theorem

81 4.5 Existence and construction of h-sets 4 F. G. W. F. Theorem (Sufficient condition) Let K be any h 1 -set contained in R see Definition 2.5.6, and consider h H (and we may relax the monotonicity condition) satisfying c 1 δ(h 1 (δr)/h 1 (r)) 1 ε h(δr)/h(r) c 2 δ(h 1 (δr)/h 1 (r)) ε for all 0 < δ, r 1, for some ε > 0. Then, there is a function W h : K R, satisfying W h (y) W h (x) c h 1 ( )/h( ) for all x, y K with 0 < 1 (as above, we consider := y x ), such that the graph Γ(W h ) is a h-set. Remark: In particular, let K R be a d 1 -set, 0 < d 1 1, and consider 1 < d < d Then, there is a function W d satisfying W d (y) W d (x) c d1+1 d, the graph of which is a d-set. In particular, by remark (a) following Definition we have dim H Γ(W d ) = dim B Γ(W d ) = dim B Γ(W d ) = d. Lemma Consider 0 < d n and let K R n be a (compact) d-set. Then, for all j N 0, there exists a cover of K by j 2 jd dyadic cubes with diameter 2 j. Proof. For all j N 0, let us to consider a covering C j of K by dyadic cubes with diameter 2 j, such that each element (cube) of the covering intersects K. Then the relation µ(k) j (#C j ) 2 jd holds, where µ is a mass distribution supported on K according to Definition 2.5.5, and therefore the Lemma follows by elementary arguments. Corollary Let 0 < s < d 1 and suppose that K R is a d 1 -set, where 0 < d 1 1. Then the maximal value of Hausdorff, lower and upper box dimensions of the graphs over all real functions f : K R belonging to C s (K), i.e. satisfying f(y) f(x) c s for all x, y K with 0 < 1, is given by d s. Proof. By remark of Theorem with d := d 1 +1 s and by remark (a) of Definition 2.5.3, it is sufficient to prove that dim B Γ(f) d s for all such f. Actually, we will prove this last inequality for all 0 < s < 1. In fact, for each ν N 0, by Lemma there are coverings of K by M(ν, K) ν 2 νd 1 dyadic intervals I k := I ν,k := [(k 1)2 ν, k2 ν ], k Z. On the other hand, since osc Ik (f) c2 νs then it holds M(ν, f) cm(ν, K) 2 νs 2 ν ν 2 ν(d 1+1 s), where M(ν, f) stands for the minimum number 72

82 4.5 Existence and construction of h-sets 4 F. G. W. F. of dyadic squares needed to cover Γ(f), and osc Ik f = sup Ik f inf Ik f is the oscillation of f over the interval I k. In this way we obtain lim ν log 2 M(ν,f) ν therefore the Proof is complete. d s, and Remark: We can easily extend the Corollary to any h 1 -set contained in R, as soon as h 1 satisfies h 1 (r) r r d r with lim r 0 + d r = d 1. This follows by the same Proof with appropriate adaptations, in particular we use the Theorem for h(r) := r 1 s h 1 (r) (instead of respective remark for d := d s, i.e. instead of h(r) := r d = r d 1+1 s ). Corollary Let 0 < s < d 1 and suppose that K R is a d 1 -set, where 0 < d 1 1. Then, the maximal Hausdorff dimension of the graphs over all real functions f : K R belonging to B s p,q(k), which are traces of continuous functions, is given by d s, for any 1 < p <. Proof. Let ε > 0 satisfy 1+ε < p. By [33], p. 165, (20.40), with Definition (20.3)/(20.4) of p. 159, we have the inclusion B s p, (K) B s 1+ε, (K). Let f B s p, (K) be a trace of a continuous real function, then f belongs to B s 1+ε, (K) and is a trace given as f = tr K g where g B s+ 1 d 1 1+ε 1+ε, (R), see [34], pp. 151 and 153. We may assume that g is real by taking the real part of such a function. In fact, according to Definition and remark (c) following it, the quasi-norms of the classic Besov spaces can be characterized by an appropriate integration and summation of pieces ϕ j f. Moreover, as we assumed that g is a continuous function on R, then by a result given in [6], step 2, p. 55, legitimated here for these spaces by elementary embeddings, it follows that the trace tr K g is indeed the restriction of the function g to the set K. In other words, it holds f = tr K g = g K. We have 0 < s + 1 d 1 1+ε < 1 because 0 < s < d 1, so by Theorem we obtain dim H Γ(g) 2 (s+ 1 d 1 ) = d 1+ε 1 +1 s+ε 1 d 1. On the other hand Γ(f) Γ(g) and so 1+ε dim H Γ(f) d s, therefore the inclusion C s (K) B s p, (K) and the Corollary complete the Proof. 73

83 5 Longer Proofs 5 Longer proofs and explanations The sequencing of the proofs in Sections below is almost entirely according the coming of the results, i.e. in general each Proof only make use of results that are proved above it in the text. Many of the Proofs are only outlined, in order to save space and to focus in the main ideas, particularly if we are applying arguments we already used before. 5.1 Proofs of Section 3.2 and Theorem Definition We recall Definition 2.2.3, and define temporarily ϕ(ν, f) := Osc(ν, f)/n ν(n 1), as well as the quantity ϕ + (ν, f) := max{1, ϕ(ν, f)}. Lemma Let f : [0, 1] n R be a continuous function. Then we have dim B Γ(f) = n + lim ν log ϕ + (ν,f) log N ν. Moreover, it holds also dim B Γ(f) 1 + lim ν log Osc(ν,f) log N ν ; in particular, if ϕ(ν, f) 1 for infinite many values of ν then this inequality is actually an equality. Proof of Lemma We have N ν (M(ν, f I ) 2) < osc I (f) N ν M(ν, f I ) for I as in Definition recall Definition Consequently, by summing over all I as in Definition we get the inequalities N ν M(ν, f) 2N ν(n 1) < Osc(ν, f) N ν M(ν, f), and by multiplying these inequalities by N ν(n 1) it follows that N νn M(ν, f) 2 < ϕ(ν, f) N νn M(ν, f). So it holds M(ν, f) < N νn (ϕ(ν, f) + 2) and M(ν, f) N νn ϕ(ν, f), and as M(ν, f) N νn is true then it holds also M(ν, f) N νn ϕ + (ν, f). Therefore we have N νn ϕ + (ν, f) M(ν, f) < N νn (ϕ(ν, f) + 2) and consequently n + log ϕ+ (ν,f) log N ν log M(ν,f) log N ν < n + log(ϕ(ν,f)+2) log N ν n + log(3ϕ+ (ν,f)) log N ν. Finally we apply lim ν and by Definition we obtain the desired equality. 74

84 5.1 Proofs of Section 3.2 and Theorem Longer Proofs Remark (a) With the same Proof, the Lemma is also true for lim ν and dim B in place of the upper counterparts. (b) Of course, Lemma and previous part (a) are immediately extendable to functions f : [ k, k] n R, with k N, taking into account the oscillations over all cubes I [ k, k] n, I as in Definition (c) Let f : [0, 1] n R be a general function. Then by arguments used in Proof of the log ϕ Lemma we have dim B Γ(f) n + lim + (ν,f) ν. If ϕ(ν, f) 1 for infinite log N ν log Osc(ν,f) many values of ν then dim B Γ(f) 1 + lim ν. Moreover, the counterparts log N ν for parts (a) and (b) of the present Remark also hold. Proof of Theorem We prove two of the four implications; the other two follow immediately. We will take into account the following equivalences, with k R and ν > 0: log Osc(ν,f) log N ν k log Osc(ν, f) k log N ν Osc(ν, f) N νk. (i) We use the inequality of Lemma Let 0 < γ 1 and dim B Γ(f) n + 1 γ. For every ε > 0, ν ε N 0 such that log Osc(ν,f) log N ν n γ + ε, ν ν ε. So Osc(ν, f) N ν(n γ+ε), ν ν ε and then we have f V γ ε (T ). Hence f V γ (T ). (ii) We use the second equality of Lemma Let 0 γ < 1 and dim B Γ(f) n + 1 γ. Then (k ν ) ν N N with lim ν k ν = such that lim ν log Osc(k ν,f) log N k ν n γ. For every ε > 0, ν ε N 0 such that log Osc(k ν,f) log N kν n γ ε, ν ν ε. So Osc(k ν, f) N k ν(n γ ε), ν ν ε and then f V γ+2ε (T ). Hence f V γ+ (T ). Proof of Remark To prove parts (a),(b) we need only replace, in the Proof of Theorem 3.2.1, the Lemma by Remark (a),(b) respectively. We still have to prove part (c). (i) The two implications of Remark (c) are equivalent, so we can commute between them in a practical and convenient way. Moreover, we prove only the upper case, as for the lower case the proof is similar. 75

85 5.1 Proofs of Section 3.2 and Theorem Longer Proofs (ii) Let f : [0, 1] n R be a general function and suppose that dim B Γ(f) n + 1 γ with 0 γ < 1. If in part (ii) of the Proof of Theorem we replace the second equality of Lemma by the second inequality of Remark (c) then we conclude that f V γ+ (T ). (iii) Let f : R n R be a general function and suppose that f V γ (R n ) with 0 < γ 1. Then, with abuse of notation we have f [ k,k] n V γ (R n ) if k N, considering only the oscillations over the cubes I [ k, k] n. By part (i) and an immediate generalization of (ii) we have dim B Γ(f [ k,k] n) n + 1 γ, and therefore the remark (c) of Definition completes the Proof. Definition Let U be a subset of R n and f : R n R a function. We define the oscillation of f over U by osc U (f) := sup U f inf U f, generalizing Definition (a). Lemma For all x := (x 1, x 2, y 1, y 2 ) R 4, consider C x := {x i y j : i = 1, 2 and j = 1, 2} and let Φ, Ψ : R 4 R be defined by Φ(x) := max C x and Ψ(x) := min C x, respectively. Then (Φ Ψ)(x) max i=1,2 x i y 1 y 2 + max i=1,2 y i x 1 x 2. Lemma Let = A R n, consider two functions f, g : A R and define ß := (sup A (f), inf A (f), sup A (g), inf A (g)). Then, by stipulating 0 = 0 = 0, we have sup A (fg) Φ(ß) and inf A (fg) Ψ(ß). Hence by Lemma we get the inequality osc A (fg) sup A f osc A (g) + sup g osc A (f). A The Proofs of Lemmas and use only standard arguments, so we shall omit them. Proof of Theorem (a) By remark (d) of Definition we may assume f, η : [0, 1] n R with f continuous and η Lipschitz. For all I [0, 1] n as in Definition 2.2.1, the Lemma gives 76

86 5.1 Proofs of Section 3.2 and Theorem Longer Proofs osc I (ηf) c osc I (f) + sup f osc I (η). Let 0 < α 1. By summing over all I [0, 1] n as in Definition 2.2.1, multiplying by N ν(α n) and taking the supremum over all ν N 0 we have ηf V α (T ) c f V α (T ) + c sup f. Then the first equivalence of Theorem gives the desired inequality. (b) By remark (c) of Definition we may assume f, η : [ k, k] n R, k N, with f continuous and η Lipschitz. In a similar way to part (a) above we get, with abuse of notation, the inequality ηf V α (R n ) c f V α (R n ) + c sup f for 0 < α 1, considering only the oscillations over the cubes I [ k, k] n. Then, by a direct generalization of the Remark (a), with [ k, k] n in place of [0, 1] n, we have the desired inequality. Proof of Theorem We prove the case 1 γ <, since for γ = the proof is analogous but simpler. (i) Consider 0 < a 1 and β k := [ λ 1 a k λ a k+1], k N. First we calculate the sum Osc(β k, W s ) of the oscillations of W s over all dyadic intervals I [0, 1] with I = 2 β k. We have the inequality Osc(β k, W s ) := I =2 β k osc I (W s ) j 1 2 λ js I =2 β k osc I ( ζ(2 λ j +θ j ) ). Because ζ is Lipschitz and bounded, the inner sum can be estimated from above by c2 min{λ j,β k }. Hence Osc(β k, W s ) can be evaluated from above by c k j=1 2 λjs 2 λ j + c j>k 2 λjs 2 β k ck2 λ k (1 s) + c2 β k j>k 2 λjs, where the estimation of the first sum is justified by the assumption 0 < s 1. Let 0 < ε < γ. There exists some strictly increasing sequence (l k ) k N N such that β lk λ lk (γ ε) a and λ l k +1 β lk (γ ε) 1 a for all k N. Since λ j+1 λ j c > 0 for j N, we have j>k 2 λ js c2 λ k+1s and then β lk Osc(β lk, W s ) cl k 2 (γ ε) a (1 s) + c 2 β l k 2 λ l +1s k cl k 2 β l k (γ ε) a + c 2 β l k (1 s(γ ε) 1 a). 1 s (ii) Let N(β lk, W s ) be the minimum number of dyadic squares, with volume 2 2β l k, required to cover Γ(W s ). Hence we have the estimate N(β lk, W s ) (Osc(β lk, W s ) + 2) 2 β l k. 77

87 5.2 Proofs of Section Longer Proofs log On the other hand, we have also dim B Γ(W s ) lim 2 N(β lk,w s ) k β lk and therefore { log dim B Γ(W s ) 1 + max 0, lim 2 Osc(β lk,w s) k β lk }. By applying the last estimate of (i), { 1 s the (inner) lower limit can be estimated from above by max, 1 s(γ ε) }, 1 a (γ ε) { a 1 s therefore we have dim B Γ(W s ) 1 + max, 1 s(γ ε) }. 1 a By choosing 0 < (γ ε) a a 1 such that γ a = 1 s + sγ we have 1 s = 1 sγ 1 a = 1 s. Therefore, the γ a 1 s+sγ Proof is now complete. Remark: If in the calculations of this Proof we take into account all values of a, instead of choosing a particular one, then we obtain also the inequality dim B Γ(W s ) 2 s. 5.2 Proofs of Section 3.3 In this Section we prove Theorems and by using a wavelet-approach, thus we present firstly the following preliminary Lemmas we will need. (Later on, in Section 6.3 below we will give other different proofs in order to compare different approaches.) Lemma ([31], p. 129) s 0 <. If s 0 n p 0 = s 1 n p 1 then we have the embedding Let 0 < p 0 p 1, 0 < q and < s 1 B s 0 p 0,q(R n ) B s 1 p 1,q(R n ). Definition ([37], p. 193) Let r N. Define L 0 := 1 and L := L j := 2 n 1 if j N. Then, there are real compactly supported functions ψ 0 C r (R n ) and ψ l C r (R n ), l = 1,..., L, with R n x α ψ l (x)dx = 0, α N n 0, α r, such that {2 j n 2 ψ l jm : j N 0, 1 l L j, m Z n } is an orthonormal basis in L 2 (R n ) where, by definition, ψ ψjm(x) l 0 (x m), if j = 0, m Z n, l = 1 :=. ψ l (2 j 1 x m), if j N, m Z n, 1 l L Lemma ([37], p. 194, particular case with elementary adaptations) Let 0 < p and s R, and take into account Definition There is a natural 78

88 5.2 Proofs of Section Longer Proofs number r(s, p) with the following property: If r(s, p) < r N and f S (R n ) then we have as an equivalent quasi-norm on B s p,1(r n ) the quantity j N 0,1 l L j 2 j(s n p +n) ( m Z n < f, ψ l jm > p ) 1 p, where < f, ψ l jm > is appropriately interpreted in (S (R n ), C r (R n )) in the natural way. The following Lemma concerns continuous real functions and is a particular case of Lemma and Formula (2) of the respective Proof. (Cf. Lemma below.) Lemma Consider α R, let f : R n R be a continuous function f S (R n ) and suppose that f = j 0 f j with convergence in S (R n ) with the weak topology, where (f j ) j N is a family of continuous complex valued functions on R n. Then, it holds (as usually, the notation Re stands for the real part of a complex valued function) f V α (R n ) Re(f j ) V α (R n ). j 0 We can see this inequality as a consequence of the following one, where I is a cube in R n as in Definition 2.2.2: osc I (f) osc I Re(f j ). j 0 Definition As an extension of Section 2.2, we define also the spaces V α (R n ) and V α (R n ), in the first case replacing osc by ess osc := ess sup ess inf and in second case by restricting the sum to the cubes I see Definition where f is continuous. Of course, we have the inequalities. V α (R n ). V α (R n ). V α (R n ). Lemma Consider α R, let f : R n R with f S (R n ), and suppose that f = j 0 f j with convergence in S (R n ) with the weak topology, where (f j ) j N is a family of continuous complex valued functions on R n. Then it holds the (triangle) inequality f V α (R n ) j 0 Re(f j ) V α (R n ). Proof of Lemma Let I be a cube in R n as in Definition 2.2.2, and consider g k := k j=1 f j and h k := 79

89 5.2 Proofs of Section Longer Proofs k j=0 sup I Re(f j) for all k N. Analogously, we define h k by using the infimum instead. We may assume h k > and h k < for all k N, otherwise the right hand side of the inequality would be and there would be nothing to prove. Consider ε > 0. Then, there exists an order k ε N such that it holds the inequality ess sup f h k + 4ε, k k ε. (1) I In order to prove this formula, suppose that (k j ) j N N with lim j k j = and satisfying ess sup I f h kj + 4ε, j N. Hence, there exists a set A I with λ(a) > 0 (strictly positive Lebesgue measure) and f(x) h kj + 3ε, x A, j N. Let S + (R n ) := {ψ S(R n ) \ {0} satisfying ψ(x) 0, x R n }. By [9], p. 310, we know that f is continuous at some a A, hence ψ S + (R n ) with f(x) h kj +2ε, x supp ψ, j N. Then in (S (R n ), S(R n )), for all j N we have < f, ψ > < h kj, ψ > +2ε ψ L1 (R n ). Hence by the convergence f = lim j g j with the weak topology in S (R n ), there exists j 0 N such that < Re(g kj ), ψ > < h kj, ψ > +ε ψ L1 (R n ) for all j j 0. This contradiction proves (1). By taking into account the Formula (1), as well as the respective counterpart for the infimum, ess inf I f h k 4ε, k k ε for some k ε N, we obtain also the inequality ess osc I (f) j 0 osc I Re(f j ). (2) Taking I =N νn, multiplying by N ν(α n) and taking the sup ν 0 we get the desired triangle inequality. Remark: As we can see by the arguments we used in the Proof above, the Formula (2) remains true if we replace I by any set A R n with strictly positive Lebesgue measure. Proof of Theorem (Wavelet approach) Analogously to the identity given in Lemma below, the reference [37], p. 194, gives the convergence f = j 0 f j in S (R n ) with the strong topology (since by [25], p. 80

90 5.2 Proofs of Section Longer Proofs 10 and (2.2) of p. 21, with reference to [3, 24], we have actually uniform convergence), where f j := l,m 2jn σ jlm ψ l jm for all j N 0 and σ jlm := < f, ψ l jm >, see Definition Hence, by recalling Lemma we know that osc I f j 0 osc I (f j) holds for all I as in Definition By [31], p. 131, f is continuous. Let 0 < p 1 and γ n. By Lemma we have p f γ+n n c f B p 1,1 (R n B γ ) p,1 (Rn ) and since γ + n n n p 1 1 p <. and γ 1 then we can assume We will take into account the Lemma with Definition Let be a general sum over all dyadics I for which we have a non-empty intersection with B R (0), and consider 1 p < and γ n p. (i) Consider ν N 0, j N 0, and I = 2 νn, I as in Definition with N = 2. (i1) Let ν j. We have osc I (f j ) 2 sup I f j. By the Hölder s inequality we obtain I =2 νn sup I f j (2 ν 2 ([R] + 1)) n p 1 p ( I =2 sup νn I f j p) 1 p, ( p 1 νn and because ν j then the right hand side is smaller than c R 2 p I =2 sup jn I f j p) 1 p. Let K 0, K 1 N be satisfying supp ψ 0 B K0 (0) and supp ψ l B K1 (0) for all l, and define K := max{2k 0, K 1 }. Thus, that value can be estimated from above by ( p 1 νn c R 2 p (2K) ) 1 n l,m 2jnp σ p p jlm c p 1 ( ) R2νn p 2 jn sup l m Z σ p 1 p n jlm. Because ν j and γ n, we have ν(γ n) j(γ n ), therefore we obtain p p p 2 ν(γ n) I =2 osc νn I (f j ) c R 2 j(γ n p +n) ( ) l m Z σ p 1 p n jlm. (i2) Let ν j. By the mean value Theorem, it holds osc I (f j ) 2 ν n i=1 sup I f j x i. Let i {1,..., n}. On the other hand, by the Hölder s inequality we obtain ( I =2 sup νn I f j x i (2 ν (2 ([R] + 1)) n p 1 1 p I =2 sup νn I f j p) p x i. Because ν j then this last expression can be estimated from above by ( p 1 νn c R 2 p 2 (ν j) n 1 p I =2 sup jn I f j p) p x i p 1 ( νn c R 2 p 2 (ν j) n p (2K) n sup l m Z (c2 j ) p (2 jn ) p ) σ p 1 n p jlm c R 2νn 2 j n p 2 jn 2 ( ) j l m Z σ p 1 p n jlm. 81

91 5.2 Proofs of Section Longer Proofs Also because ν j, we have ν min{1, γ} ν + j j min{1, γ}. So, it holds 2 ν(min{1,γ} n) I =2 osc νn I (f j ) c R 2 j(min{1,γ} n p +n) ( ) l m Z σ p 1 p n jlm. (i3) Looking at (i1) and (i2), in particular we get for all j, ν N 0 the inequality (ii) Let ν N 0. 2 ν(min{1,γ} n) I =2 osc νn I (f j ) c R 2 j(γ n p +n) ( ) l m Z σ p 1 p n jlm. By using the inequality osc I (f) j 0 osc I (f j) and by taking into account Lemma we obtain 2 ν(min{1,γ} n) I =2 νn osc I (f) c R f B γ p,1 (Rn ). Of course, we may replace by in the obtained inequality. Then by taking the sup ν 0 on the left hand side we have the desired inequality. Proof of Corollary By [31], p. 131, f is continuous. Let η C (R n ) with supp η [ 2, 2] n and η(x) = 1 if x [ 1, 1] n. By remark (c) of Definition and by Theorem (a) we have dim B Γ(f) = sup k N dim B Γ(η( /k)f). Let k N. By Lemma we have η( /k)f n p σ<γ Bσ p,1(r n ). If 0 < p < then by Theorem we have η( /k)f V min{1,γ} (R n ). On the other hand, if p = then the identity B γ (R n ) = C γ (R n ) and a generalization of the remark (c) of Definition give η( /k)f V min{1,γ} (R n ). Hence Remark (b) gives dim B Γ(η( /k)f) n + 1 min{1, γ}. Lemma ([37], p. 194, particular case with elementary adaptations) Let 0 < p and s R, and take into account Definition There is a natural number r(s, p) with the following property: If r(s, p) < r N and f S (R n ) then we have as an equivalent quasi-norm on B s p, (R n ) the quantity sup j N0,1 l L j 2 j(s n p +n) ( m Z n < f, ψ l jm > p ) 1 p, where < f, ψ l jm > is appropriately interpreted in (S (R n ), C r (R n )) in the natural way. Proof of Theorem (Wavelet approach) Because f V γ max{1,p} (R n ), of course f is bounded and measurable, and therefore 82

92 5.2 Proofs of Section Longer Proofs f S (R n ). We will take into account the Lemma with Definition (i) Consider 0 < p 1 and j N 0, and let K 0 > 0 be such that supp ψ 0 B K0 (0), and K 1 > 0 be such that supp ψ l B K1 (0) for all l with 1 l L. Defining K := max{k 0, K 1 }, we have < f, ψ l jm >= 0 for all j N 0, 1 l L j, m Z n with m > m R := [2 max{1,j} 1 (R + K)] + 1. Analogously to part (i) of the Proof of Theorem 6.3.7, by Hölder s inequality we can estimate < m m R f, ψ l jm > p from above by ( ( < f, ψ l m mr jm > p ) 1 ) p ( p ) 1 p. m m p R Hence f B γ p, (R n ) c R f B γ 1, (Rn ), and therefore we may assume 1 p <. (ii) Consider now the case 1 p < and let f L p (R n ) V γp (R n ). We will first analise separately in (ii1) the contribution of the term with j = 0. On the other hand, concerning the terms with j N, we will calculate in (ii2) their contribution for the oscillation of the function. (ii1) The term originated from j = 0 is estimated by the elementary inequality ( m Z n < f, ψ 0 ( m) > p) 1 p c f Lp(R n ). (ii2) We notice here that, for j N, 1 l L, m Z n, I = 2 j n j (where = j 1), I as in Definition 2.2.2, the hypothesis R n ψ l (x)dx = 0 justifies that see Definition (ii1/ii2) with K 1 in place of 2 jε the identity < f, ψ l jm > = t 2 j K 1 ψ l (2 j t)f(2 j m + t)dt = t 2 j K 1 ψ l (2 j t)f I (2 j m + t)dt holds and then, whenever 2 j m I, we have < f, ψ l jm > t 2 j K 1 ψ l (2 j t)f I (2 j m + t) dt. Thus, for 2 j m I it holds < f, ψ l jm > c2 j n K n 1 sup I # f I, and therefore 2 j( n p +n) ( m Z n < f, ψ l jm > p ) 1 p c2 j( n p +n) ( I =2 j n = c ( ( 2 j n sup I # f I ) p ) 1 p 2 ) 1 j n I =2 j n sup I # f I p p. For the rest of the Proof we apply the same calculation techniques as in part (ii) of the Proof of Theorem 6.3.7, putting γ = γ, j N 0 in place of j N, and K 1 in place of 83

93 5.3 Proofs of Section Longer Proofs 2 jε. In this way we get 2 j(γ n p +n) ( m Z n < f, ψ l jm > p ) 1 p c f V γp (R n ). So, we get sup j N,1 l L 2 j(γ n p +n) ( m Z n < f, ψ l jm > p ) 1 p c f V γp (R n ). (ii3) By (ii1) and (ii2), taking into account Lemma 5.2.5, we obtain the inequality f B γ p, (R n ) c max { f Lp(R n ), f V γp (R n )}, in other words f B γ p, (R n ) c f Lp(R n ) V γp (R n ). The Proof is now complete. 5.3 Proofs of Section 3.4 We start by stating the following Lemma 5.3.1, which is implicitly used for the existence of the functions φ as in Definitions 3.4.1, and or as in Theorem The mollification techiques used in the Proof of this Lemma are also useful in order to define the function F ϕ on [1/2, 1] as in remark (a) of Definition Lemma Consider g C (R n ) and K a compact set in R n. Then, there exists h C0 (R n ) satisfying h(x) = g(x), x K. Proof of Lemma Let ε > 0 and K ε := {x R n with d(x, K) < 2ε}, where d is the usual distance between points and/or sets in R n. Consider f : R n R with f(x) := 1 if x K ε and f(x) := 0 otherwise, and let f ε be the corresponding mollified function (see [32], p. 41). By [32], pp , we have f ε C0 (R n ), so by taking h := f ε g the Proof is complete. Lemma Let r > 1. Then, by the mean value Theorem, for some c (r, r + 1) 1 we have = 1. Hence for m N with m 2 we have m j=2 1 ln(r+1) ln r 1 c ln 2 c (r+1) ln 2 (r+1) 1 2 j 1 ln(m+1) 1 ln j 1 j 2 2 j 1 ln(j+1) 1 ln j (j+1) ln 2 (j+1) j 2 <. 2 j Proof of Theorem (I) For n = 1: 84

94 5.3 Proofs of Section Longer Proofs (i) We prove that Θ is bounded on R and continuous on R \ {0}: We have x ψ(x) c, x R. (i1) Let x 0 and j 1 N \ {1, 2}. Then j j 1 Θ j (x) = j j 1 ψ (2 j+j 0 ( )) x 1 ln j j j 1 c 2 j+j 0 x 1 ln j j j 1 c 2 j ln j. Hence j j 1 Θ j is bounded on (, 0] and converges uniformly on (, 0]. In particular Θ is continuous on (, 0). (i2) Let j 1 N \ {1, 2}. Let x 1 ln j 1. Then j j 1 +1 Θ j(x) = ( ( )) j j 1 +1 ψ 2 j+j 0 x 1 ln j Hence [ ) j 3 Θ 1 j converges uniformly on ln j 1,. In particular Θ is continuous on (0, ). (i3) Let x > 0. Let j(x) := min{j N \ {1, 2} with x 1 ln j }. We have j j 1 +1 j 3 Θ j(x) 2 sup ψ + j(x) 2 j=3 Θ j (x) + j j(x)+1 Θ j(x) 2 sup ψ + j(x) 2 j=3 c 2 j+j 0 1 ln(j(x) 1) 1 ln j + j j(x)+1 c 2 j+j 0 1 ln j(x) 1 ln j. The second sum on the right hand side can be estimated from above by and the factor before the sum tends to 0 when x 0 +. On the other hand by Lemma we have j(x) 2 j=3 Hence we have j 3 Θ j(x) 2 sup ψ + c2 j 0, x > 0. Therefore Θ is bounded on (0, ). c 2 j+j 0 x 1 ln j c j 1 j j j. 2 j(x) 1 ln j(x) 1 ln(j(x)+1) 1 2 j 1 ln(j(x) 1) 1 ln j c <. j 1 c, 2 j+j 0 (ii) We may assume ψ(0) = 18. For the general case, in what follows we need only change some constants by the factor ψ(0)/18 0. Let j 1 N \ {1, 2} and j 0 sufficiently large. ( ( 1 1 We have Θ = Θ j1 + ( j 3 j j 1 Θ j ln j 1 ) ln j 1 ) ) 1 ln j 1 ψ(0) j 3 j j 1 Θj ( ) 1. ln j 1 By calculations like (i3) the last sum can be evaluated from above by c2 j 0 1. Then 85

95 5.3 Proofs of Section Longer Proofs ( ) 1 Θ ln j Also by calculations like in (i3) we have Θ ( ln j 1 ln(j 1 +1) 2 ) c2 j 0 1. Therefore Let 0 < ε < 1. sup [ 1 x,y ln(j 1 +1), 1 ln j 1 ] Θ 0(x) Θ 0 (y) 17 1 = 24. ln j 1 ln j 1 ln j 1 For every ν N 0 let k(ν) := [N ν(1 ε) ]. Then k(ν) 1 2 N ν(1 ε), ν N 0. Let ν 0 N 0 such that 2e N νε N ν(1 2ε) 1, ν ν 0. Then ( ) 1 ln t 1 1 N ν 1 if k(ν)n ν with t 3 and ν ν 2 ln t 0. Here ( ) represents the derivative operator. Let ν 1 ν 0 with k(ν 1 )N ν 1 1 ln 3. If ν ν 1 then for every k {1,..., k(ν)} there exists j ν,k N \ {1, 2} such that { } 1 1, ln j ν,k ln(j ν,k + 1) Hence [ (k 1)N ν, kn ν]. sup Θ 0 (x) Θ 0 (y) (k 1)N ν, k {1,..., k(ν)}, ν ν 1. x,y [(k 1)N ν,kn ν ] ln j ν,k Let I as in Definition We have k(ν) osc I (Θ 0 ) 2 4 (k 1)N ν, ν ν 1. I =N ν k=1 Then I =N ν osc I (Θ 0 ) 2 4 (k(ν) 1)k(ν) 2 N ν, ν ν 1. For ν ν 2 with ν 2 ν 1 sufficiently large we have Osc(ν, Θ 0 [0,1] ) 2 2 (k(ν)) 2 N ν N ν(1 2ε). Hence, if we take 0 < 2ε < 1 and use the lower counterpart of the second equality of Lemma (according to Remark (a)) then we have dim B Γ(Θ 0 [0,1] ) 2 2ε. Hence dim B Γ(Θ 0 [0,1] ) = 2 and, by using similar but simpler arguments, we have also dim B Γ(Θ [0,1] ) = 2. 86

96 5.3 Proofs of Section Longer Proofs (iii) Here we choose a > 0. Let ϕ be as in Definition 2.1.8, and with supp(f ϕ) {ξ R n : 1 (1 + a) ξ 2(1 a)} and (F ϕ)(ξ) = 1 if 1 a ξ 1 + a, for some a > 0. 2 Let s < 1 p and 0 < q. In the next first equality we make use of the continuity ([22], p. 18) of the convolution operator on S (R). We have ( ) j 0 2 js q n ( ϕ j Θ Lp (R) = (2π) 2 j 3 2 (j+j 0 )s q Θ j Lp (R)) = ( ) = (2π) n 2 2 (j+j0)s 2 (j+j 0) 1 q p ψ Lp (R) = c j 3 2(j+j 0)(s p)q 1 <. (Standard j 3 modification if q =.) Hence the relation Θ B s p,q(r) holds. By Lemma we have also Θ 0 B s p,q(r). (II) For n N \ {1}: This part of the Proof is analogous to part (I), although with more fastidious calculations. We make a sketch. In part (i) (respectively (ii)) we use essentially the fact that the corresponding part in (I) remains valid if we replace Θ by Θ(, x 2,..., x n ), for all (respectively a convenient lattice of) fixed points (x 2,..., x n ) R n 1. (i) We prove that Θ is bounded on R n and continuous for x 1 R \ {0}: We have x ψ(x) c, x R n. (i1) Let x 1 0 and j 1 N \ {1, 2}. Then j j 1 j 2(n 1) l=1 Θ j,l (x) j j 1 cj 2(n 1) 2 j ln j. Hence j j 1 j 2(n 1) l=1 Θ j,l is bounded for x 1 (, 0] and converges uniformly for x 1 (, 0]. In particular Θ is continuous for x 1 (, 0). (i2) Let j 1 N \ {1, 2}. Let x 1 1 ln j 1. Then j j 1 +1 j 2(n 1) l=1 Θ j,l (x) c j1 j j 1 +1 Hence j 2(n 1) j 3 l=1 Θ j,l converges uniformly for x 1 j 2(n 1) 2 j. [ 1 ln j 1, ). 87

97 5.3 Proofs of Section Longer Proofs In particular Θ is continuous for x 1 (0, ). (i3) Let x 1 > 0. Then we have j 3 j 2(n 1) l=1 Θ j,l (x) 2 n sup ψ + c2 j 0. Therefore Θ is bounded for x 1 (0, ). (ii) We may assume ψ(0) = 18. Let j 1 N \ {1, 2} and l 1 {1,..., j 2(n 1) } with the corresponding (s 2,l1,..., s n,l1 ) {1,..., j 2 } n 1. Let j 0 sufficiently large. By calculations like in (i3) we have ( 1 Θ, s 2,l 1,..., s ) ( n,l 1 1 = Θ ln j 1 j1 2 j1 2 j1,l 1, s 2,l 1 ln j 1 j1 2 ( 1 Also we have Θ 1 + ln j 1 ln(j 1 +1), s 2,l 1 2 j1 2 Therefore by fixing (x 2,..., x n ) = ( s2,l1 j 2 1 Let 0 < ε < 1. sup [ 1 x 1,y 1 ln(j 1 +1), 1,..., s n,l 1 j 2 1 ψ(0) c2 j ),..., s n,l 1 c2 j 0 1. j1 2 ),..., s n,l 1 we have j1 2 ln j 1 ] Θ 0(x) Θ 0 (y) ) + (j,l) (j 1,l 1 ) 17 1 = 24. ln j 1 ln j 1 ln j 1 For every ν N 0 let k(ν) := [N ν(1 ε) ]. Then k(ν) 1 2 N ν(1 ε), ν N 0. Let ν 0 N 0 such that 2e N νε N ν(1 2ε) 1, ν ν 0. Then ( ) 1 ln t 1 1 N ν 1 if k(ν)n ν with t 3 and ν ν 2 ln t 0. ( 1 Θ j,l, s 2,l 1 ln j 1 j 2 1,..., s n,l 1 j 2 1 Let ν 1 ν 0 with k(ν 1 )N ν 1 1 ln 3. If ν ν 1 then for every k {1,..., k(ν)} there exists j ν,k N \ {1, 2} such that { } 1 1, ln j ν,k ln(j ν,k + 1) Hence by fixing (x 2,..., x n ) = ( s2,l1 j 2 1,..., s n,l 1 j 2 1 [ (k 1)N ν, kn ν]. ) we have sup Θ 0 (x) Θ 0 (y) (k 1)N ν, k {1,..., k(ν)}, ν ν 1. x 1,y 1 [(k 1)N ν,kn ν ] ln j ν,k 88 )

98 5.3 Proofs of Section Longer Proofs By Lemma we have lim j1 1/j ln j 1 ln(j 1 +1) = 0. Hence if ν 2 ν 1 is sufficiently large then with I as in Definition we have k(ν) osc I (Θ 0 ) N ν(n 1) 2 4 (k 1)N ν, ν ν 2. I =N νn k=1 Then I =N νn osc I (Θ 0 ) 2 4 (k(ν) 1)k(ν) 2 N ν(n 2), ν ν 2. For ν ν 3 with ν 3 ν 2 sufficiently large we have Osc(ν, Θ 0 [0,1] n) 2 2 (k(ν)) 2 N ν(n 2) N ν(n 2ε). Hence, if we take 0 < 2ε < 1 and use the lower counterpart of the second equality of Lemma (according to Remark (a)) then we have dim B Γ(Θ 0 [0,1] n) n+1 2ε. Hence dim B Γ(Θ 0 [0,1] n) = n + 1 and of course we have also dim B Γ(Θ [0,1] n) = n + 1. (iii) Here we choose a > 0 and ϕ like in part (I)-(iii). Let s < n p and 0 < q. In the next equality we make use of the continuity ([22], p. 18) of the convolution operator on S (R n ). We have ( ) j 0 2 js q n ϕ j Θ Lp (R n ) = (2π) 2 (2π) n 2 = (2π) n 2 ( j 3 j 3 j2(n 1) q p 2 (j+j 0)s j 2(n 1) 1 p Θj,1 Lp(R n )) q = ( j 3 ( 2 (j+j 0)s 2 (j+j 0) n p ψ Lp (R n )) q = 2 (j+j 0)s = c j 3 j2(n 1) q p 2 (j+j 0 )(s n p )q <. (Standard modification if q =.) q j 2(n 1) l=1 Θ j,l Lp(R )) n Hence the relation Θ B s p,q(r n ) holds. By Lemma we have also Θ 0 B s p,q(r n ). The following Remark extends and generalizes Theorem Remark (a) The Theorem is true also for all j 0 N 0 in Definition (b) In Proof of Theorem we can replace the last inequality by. Therefore we have Θ B n p p (R n ) for 0 < p. In fact, as we will see in Theorem 3.4.4, we have also Θ 0 B n p p (R n ) for 0 < p. 89

99 5.3 Proofs of Section Longer Proofs (c) If in Definition we replace j 2 by [ j δ+1], and 1 ln j by 1 j δ, with δ > 0, then Theorem remains true but with dim B Γ(Θ 0 ) = n δ instead. Proof of Remark We give a sketch of the proofs of (a) and (c), recalling techniques we used before. (i) The equality dim B Γ(Θ 0 ) = n + 1 remains true for all j 0 N 0. In fact if in (I)-(ii) and (II)-(ii) of the Proof of Theorem we replace j 1 3 by j 1 j 2 with j 2 N 0 sufficiently large we get this equality. This proves (a), so it remain us to prove (c). (ii) By remark (a) of Definition we can assume N = 2. Consider ν N 0 and define j ν N such that j ν ν 2 ν 1+δ, so 1 jν δ ν 2 ν δ 1+δ (. Because 1 ) t = δ, t > 0, and δ t t δ+1 ν 2 ν 1, then it holds 1 (j+1) δ j c2 ν if j j δ ν. So, if in parts (ii) of the Proof δ j δ+1 ν of Theorem we put k(ν) := [2 ν 1+δ ], and replace 1 by 1 and ν ln j j δ 0 by 0, then osc I(Θ) 1+δ δ ν 2 ν 2 ν δ 1+δ 2 ν(n 1) = 2 ν 1 1+δ 2 ν(n 1). ] [0,1] n 1 I =2 ν I [0,2 ν On the other hand, by calculations like in the estimation of j in part (I)-(ii3) of the Proof of Theorem (b), and by comments at the begin of part (II) of the same Proof, we obtain osc I(Θ) cj I =2 ν I [2 ν 1+δ δ ν 2 ν(n 1) = c2 ν 1,1] [0,1] n 1 1+δ 2 ν(n 1). Hence I =2 ν osc I (Θ [0,1] n) ν 2 ν 1 1+δ 2 ν(n 1) holds, therefore dim B Γ(Θ) = (n δ ) + 1 = n δ, so we have (c). Proof of Theorem We make only a layout, in order to avoid the complete fastidious calculations. For the convenience of the presentation, we assume 0 < λ < 1. For the case λ = 1 the proof is similar but with parts (i) and (ii) much more easy. We observe that x m ψ(x) c m for all m N 0 and that 2 (j+1)λ 2 jλ k2 j 2 jλ 2 j, k = 1,..., k(j), for all j N. (I) For n = 1: (i) By the same techniques used in part (i) of the Proof of Theorem 3.4.1, we can show that we have uniform convergence of the series Λ = k(j) j 1 k=1 Λ j,k in all R. Then Λ 90

100 5.3 Proofs of Section Longer Proofs is bounded and continuous. (i1) In fact, with the same techniques used in part (i1) of the Proof of Theorem 3.4.1, we show the uniform convergence of k(j) j 1 k=1 Λ j,k on (, 0]. (i2) Also, with the same techniques used in parts (i2) and (i3) of the Proof of Theorem 3.4.1, we show the uniform convergence of k(j) j 1 k=1 Λ j,k on (0, ). (ii) Let j N and ρ 0. We define ρ j as a general sum over the intervals I C ρ for all I as in Definition with I = 2 j, where C ρ := {t R : t ρ}. Then, for r N 0 and ρ 0, there is a positive constant c r independent of j and ρ such that ρ r ρ osc I (ψ) c r. (3) j Here we consider 0 0 = 1. In order to prove the basic but important Formula (3), we notice easily that the left hand side can be estimated from above by the sum I =2 j sup I t r ψ (t) 2 j I =2 j sup I ( ) (1 + t r c ) r 2 j 1+ t r + t r+2 2c r ω 0 g r(ω2 j )2 j, where g r (s) := 1+sr 1 = 1+s r +s r+2 1+s 2 (1 1 ), s 0, is a decreasing function on R+ 0. Hence 1+s r we have the inequality ω 0 g r(ω2 j )2 j g r (0)2 j + g 0 r (s)ds c r, which proves (3). We define S ρ,u,v j,u,v := ρ,u,v j osc I (Λ u,v ), u N, v {1,..., k(u)}, where the sum ρ,u,v j is taken over the intervals I C ρ,u,v for all I as in Definition with I = 2 j, where C ρ,u,v := {t R : t m(u, v) ρ}. Then by Formula (3) there is a positive constant c r independent of j, ρ, u and v such that (2 u ρ) r S ρ,u,v j,u,v c r. (4) Let S j,u,v := I =2 osc j I (Λ u,v ), u N, v {1,..., k(u)}. Then, of course we have k(u) osc I (Λ) S j,u,v. (5) I =2 j u 1 v=1 91

101 5.3 Proofs of Section Longer Proofs Let j, j and j be general sums over the intervals I C for all I as in Definition with I = 2 j, where C represents the set (, 0], [0, 2 jλ ] and [2 jλ, ), respectively. (ii1) By Formula (4) and calculations like in (i1) for x 0, but with S x m(u,v),u,v j,u,v in place of Λ u,v (x), we have the inequality k(u) u 1 v=1 S j,u,v c, where by definition S j,u,v is obtained from S j,u,v by replacing I =2 by j j. Therefore, by the counterpart of Formula (5) for j, we have the estimation j osc I (Λ) c. (ii2) Because Λ is bounded, then we have the estimation j osc I (Λ) c2j(1 λ). (ii3) Also as counterpart of Formula (5) we have the inequality j osc I (Λ) k(u) u 1 v=1 S j,u,v, where by definition S j,u,v is obtained from S j,u,v by replacing I =2 j by j. We denote by j or j the right hand side of this inequality, if we make the restriction 1 u j or u > j, respectively. Then, by Formula (4) and calculations like in (i2) for x 2 jλ, but with S x m(u,v),u,v j,u,v in place of Λ u,v (x), we obtain j c. On the other hand, as a particular case of Formula (3) with 0 in place of ρ and r, we have I =2 j osc I (ψ) c, where c is independent of j. Then we have the inequality j j u=1 k(u)c c 2 j(1 λ), and therefore the estimation j osc I (Λ) c2j(1 λ). (ii4) Hence by (ii1), (ii2) and (ii3) we have Λ V λ (R). By remark (d) of Definition and by Theorem we have dim B Γ(Λ) 2 λ. (iii) Here we choose a > 0. Let ϕ be as in Definition 2.1.8, and with supp(f ϕ) {ξ R n : 1 (1 + a) ξ 2(1 a)} and (F ϕ)(ξ) = 1 if 1 a ξ 1 + a, for some a > 0. 2 Let s R and 0 < q. In the next equality we make use of the continuity ([22], p. 18) of the convolution operator on S (R). We have (the last sum is taken over all j 1 large such that k(j) 0) q k(j) ( 2 js q n ϕ j Λ Lp (R)) = (2π) 2 2 js Λ j,k j 0 j 1 j 1 ( ) ) 1 q (2 js 1jp p k(j)2 j ψ Lp(R) 92 k=1 1 Lp(R) λ j q 2j(s p )q. j 1

102 5.3 Proofs of Section Longer Proofs (Standard modification if q =.) (II) For n N \ {1}: The layout for this part of the Proof is essentially the one used in the first part but with calculations slightly more complicated. In parts (i) and (ii) we use essentially the fact that the respective parts of (I) remain valid valid if we replace Λ by Λ(, x 2,..., x n ), for all fixed points (x 2,..., x n ) R n 1. (i) We have uniform convergence of the series Λ in all R n. Then Λ is bounded and continuous. (ii) Let j N. Similarly to the corresponding part in (I), we define j, j, j and j as general sums over the sets I C for all I as in Definition with I = 2 jn, where C represents the set (, 0] [0, 1] n 1, [0, 2 jλ ] [0, 1] n 1, [2 jλ, ) [0, 1] n 1 and R (R n 1 \ [0, 1] n 1 ), respectively. For fixed (x 2,..., x n ) R n 1, part (ii) of the case (I) is valid for Λ(, x 2,..., x n ), in place of Λ. By applying this idea, we obtain the following (ii1)-(ii3). (ii1) We have j osc I (Λ) c2j(n 1). (ii2) Because Λ is bounded, we have j osc I (Λ) c2j(1 λ) 2 j(n 1). (ii3) We have j osc I (Λ) c2j(1 λ) 2 j(n 1). (ii4) In an analogous way to (ii1), we have also j osc I (Λ) c2 j(n 1). (ii5) Hence by (ii1), (ii2), (ii3) and (ii4) we have Λ V λ (R n ). Definition and by Theorem we have dim B Γ(Λ) n + 1 λ. By remark (d) of (iii) Here we choose a > 0 and ϕ like in part (I)-(iii). Let s R and 0 < q. In the next equality we make use of the continuity ([22], p. 18) of the convolution operator on S (R n ). We have (the last sum is taken over all j 1 large such that k(j) 0) ( 2 js q k(j) n 2 ϕ j Λ Lp(R )) j(n 1) n = (2π) 2 2 js Λ j,k,l j 0 j 1 k=1 l=1 Lp (R n ) q 93

103 5.3 Proofs of Section Longer Proofs j 1 ( ) ) 1 q (2 js 1jp k(j)2j(n 1) 2 jn p ψ Lp (R n ) (Standard modification if q =.) 1 λ j q 2j(s p )q. j 1 The following Remark extends and generalizes Theorem Remark (a) Let 0 < λ < 1 and 0 < r 1 2 λ. Then the Theorem is also true with k(j) = [r2 j(1 λ) ] in Definition (b) If in Definition we replace 1 j by 1 j u the Theorem remains true. with u > 0 (u > 1 in case λ = 1), then Proof of Theorem The Proof is similar to that one of Theorem We make a layout for part (c). We need modification if h = n 1 in the case n N \ {1}. Of course for Θ in place of Θ 0 we need only to remove the factors φ and 1 ln j. We choose a > 0. Let ϕ be as in Definition 2.1.8, and with supp(f ϕ) {ξ R n : 1(1 + 2a) ξ 2(1 2a)} and (F ϕ)(ξ) = 1 if 1 2a ξ 1 + 2a, for some a > 0. 2 ( (i) For n = 1 we make use of a generalization of the Lemma with F φ ) k(j) k=1 Θ j,k = ( (2π) n 2 (F φ) F ) k(j) k=1 Θ j,k in place of the a j ζ j for all j N \ {1, 2}. Then we have a generalization of the Proof of Theorem 4.2.1, by starting in Formula (7) with φ k(j) k=1 Θ j,k in place of a j W j. Hence, for s R and 0 < q we have the equivalence (standard modification if q = ) Θ Bp,q(R) s k(j) 2 js j 3 φ Θ j,k k=1 Lp(R) On the other hand we have the relations φ ( ) 1 k(j) k=1 Θ 1 j,k j k(j)2 j p (ln j) p Lp (R) q <. j 1 ln j 2 j λ p. ( (ii) For n N\{1} we use a generalization of the Lemma with F φ k(j) ) d(j) n 1 k=1 l=1 Θ j,k,l = ( (2π) n 2 (F φ) F k(j) ) d(j) n 1 k=1 l=1 Θ j,k,l in place of the a j ζ j for all j N \ {1, 2}. Then we have a generalization of the Proof of Theorem 4.2.1, by starting in Formula 94

104 5.3 Proofs of Section Longer Proofs (7) with φ k(j) d(j) n 1 k=1 l=1 Θ j,k,l in place of a j W j. Hence, for s R and 0 < q we have the equivalence (standard modification if q = ) Θ Bp,q(R s n ) k(j) d(j) 2 js j 3 φ n 1 k=1 l=1 Θ j,k,l Lp(R n ) q <. On the other hand we have the relations φ k(j) ( ) 1 d(j) n 1 Lp 1 k=1 l=1 Θ j,k,l j k(j)d(j) n 1 2 jn p (R n (ln j) ) p h+λ 1 j 2 j p ln j. Proof of Theorem (a) (i) The Theorems and (a) give dim B Γ(φΛ) n + 1 λ. Moreover, in order to prove the second equality we can assume that supp φ is compact. This follows by Theorem and remark (c) of Definition By the same proof of part (b) we have φλ B λ p p (R n ) for 0 < p. Then, by Remark (b) with 1 in place of p (respectively remark (b) of Definition if λ = 1), we obtain dim B Γ(φΛ) n + 1 λ. (ii) By (i) we have dim B Γ(φΛ) = n + 1 λ, in particular the left hand side does not depend on φ. Then we choose a particular φ(x) 0, x R n, and apply Theorem 3.2.3, getting also dim B Γ(Λ) = dim B Γ(φΛ). (b) The proof of this part is similar to the proof of part (c) of Theorem We make a layout, and assume 0 < λ < 1, since the case λ = 1 is similar. We choose a > 0. Let ϕ be as in Definition 2.1.8, and with supp(f ϕ) {ξ R n : 1(1 + 2a) ξ 2(1 2a)} and (F ϕ)(ξ) = 1 if 1 2a ξ 1 + 2a, for some a > 0. 2 (i) For n = 1, s R and 0 < q we have the equivalence (standard modification if q = ) Λ B s p,q(r) j 1 2 js k(j) φ Λ j,k k=1 Lp (R) q <. On the other hand we have (the j are valid for j sufficiently large) φ ( ) 1 k(j) k=1 Λ 1 j,k j k(j)2 j p 1 λ j p j j 2 j p. Lp(R) 95

105 5.4 Proofs of Section 3.5 (except Theorem 3.5.3) 5 Longer Proofs (ii) For n N \ {1}, s R and 0 < q we have the equivalence (standard modification if q = ) Λ B s p,q(r n ) j 1 2 js k(j) φ k=1 2 j(n 1) l=1 Λ j,k,l Lp(R n ) q <. On the other hand we have (the j are valid for j sufficiently large) φ k(j) ( ) 1 2 j(n 1) 1 k=1 l=1 Λ j,k,l j k(j)2 Lp(R j(n 1) 2 jn p 1 λ n j ) p j j 2 j p. 5.4 Proofs of Section 3.5 (except Theorem 3.5.3) Proof of Theorem In Theorem we proved (a). The argument in [37], pp , proves not only Theorem but also the relation f s B s p (R n ) for 0 < p and s > 0. Hence, by Corollary with in place of p, and by Remark (b) with 1 in place of p, we obtain the equality dim B Γ(f s ) = n + 1 min{1, s} for s > 0. Hence it holds dim s p n + 1 min{1, s} Dim s p for 0 < p and s > 0. By Corollary we obtain (b). We still have to prove (c) and (d). 1 (i) Let 0 < p < and 0 < s. By Corollary and remark (b) of Defini- max{1,p} tion we have dim s p n + 1 s max{1, p}. On the other hand, the Theorem gives dim s p n + 1 sp for 0 < s 1. Hence we have (c). p (ii) Let Λ as in Definition if we take λ = 1 and multiply Λ j,k and Λ j,k,l by the factor 2 j( 1 p s). Let φ be as in Theorem and consider s > 1. Of course Λ is con- p tinuous, and by the same Proof of Theorem (b) we obtain φλ B s p (R n ). By calculations like in the estimation of j, in part (I)-(ii3) of the Proof of Theorem (b), and by comments at the begin of part (II) of the same Proof, we obtain dim B Γ(Λ) = n. By Theorem and by remark (b) of Definition we obtain also dim B Γ(φΛ) = n. Hence we have (d). Proposition Consider f : A R n R bounded, compactly supported, and continuous on A \ J, where dim B J n 1. As a modification of Definition 2.4.1, let 96

106 5.4 Proofs of Section 3.5 (except Theorem 3.5.3) 5 Longer Proofs M (ν, f) be obtained as M(ν, f) when we restrict the function f to the cubes I as in Definition where it is continuous. Then we have Proof of Proposition dim B Γ(f) = lim ν log M (ν,f) log N ν. Consider ε > 0. Choose ν ε N 0 in such a way that M(ν, J) N ν(n 1+ε), ν ν ε, where M(ν, J) is defined by replacing R n+1 by R n and Γ(f) by J in Definition Therefore, it holds M(ν, f) M (ν, f) 2(sup f + 1)N ν 3 n M(ν, J) cn ν(n+ε), ν ν ε. Hence, if dim B Γ(f) > n then by taking ε > 0 sufficiently small it follows that lim ν log M(ν,f) log N ν it complete the Proof. log M = lim (ν,f) ν. Finally, Definition and remark (b) following log N ν Proposition Let 0 α < 1. By restricting Definition to the functions as in Proposition 5.4.1, we obtain the identities V α (R n ) = V α (R n ) = V α (R n ). Proof of Proposition By the inequalities given in Definition 5.2.2, it is sufficient to prove, under the given hypotheses, the inclusion V α (R n ) V α (R n ). Consider f such a function, continuous on R n \ J with dim B J n 1. Let be the sum restricted to the cubes I as in Definition such that I J. Then we have I =N osc νn I (f) 2 sup f 3 n M(ν, J), ν 0. Let ε > 0. Then, there exists ν ε N 0 such that M(ν, J) N ν(n 1+ε), ν ν ε. By taking ε > 0 sufficiently small we obtain N ν(α n) I =N νn osc I (f) c sup f, ν ν ε. Proof of Remark (d) As we can see in Proof of Theorem (c), all we need to prove is that Theorem remains true for the functions f as in Proposition instead. Well, if we restrict the oscillations to the cubes I where such a function f is continuous, then Lemma remains true; this follows by replacing M(ν, f) by M (ν, f) 97

107 5.4 Proofs of Section 3.5 (except Theorem 3.5.3) 5 Longer Proofs see Definition and Proposition and using this Proposition instead of Definition in the Proof of this Lemma. So, Theorem remains true for the spaces V (R n ) if restricted to functions as in Proposition Proposition completes the Proof. Therefore, the Proof of Theorem (i) For the first table, the very difficult part ( ) is given by François Roueff in the deep Theorem 4.8 of [25], p. 77. On the other hand, parts ( ) and ( ) follow by Theorem and Remark 3.5.2, and the remaining two cells n are proved by Corollary and comments that follow the respective Proof. (ii) Concerning the second table, let us prove the first two cells n + 1. In order to do this of course we may assume s > 0 large, e.g. s n. We consider the function p Θ 0 as in Definition but replacing Θ j by a j Θ j, and analogously for Θ j,l, where a j = 1 if j is even and a j = 2 j( n p s) if j is odd. Then, the Proofs of Theorems and 3.4.4, with elementary adaptations, give dim B Γ(Θ 0 ) = n + 1 and Θ 0 B s p (R n ) for 0 < p. For the proof of the third cell n + 1 we follow the existence argument applied in the proof of Theorem 1 of [16]. We simply consider an adapted version of the argument, fitting with the dimension n + 1 and the factor a j = 2 js if j is odd introduced (analogously as above) in the trigonometric series, here multiplied by an appropriate infinitely smooth and real cutoff function in order to apply Theorem For the proof of the cell n we may assume s > 0 small, e.g. 0 < s n, and consider p the function Θ 0 as in Definition 3.4.2, which has a graph with Hausdorff dimension n. It remains to prove the cells ( ), for which we use Remark (b) and remark (b) of Definition for the part. For part, the case 1 p is proved by Theorem and respective Proof, with elementary adaptations concerning general ϕ as in Definition The case 0 < p 1 is proved by Remark (b) applied to ρ = 2. 98

108 5.5 Proofs of Section 4.2 and Theorem Longer Proofs 5.5 Proofs of Section 4.2 and Theorem Lemma Let a > 0 and e 1 := (1, 0,..., 0) R n. Then, for x R n, we have the implication x e 1 a = x e 1 C a x, with C a := inf x e 1 x infimum is taken over all x R n with x e 1 a. > 0, where the Lemma Let ϕ be as in Definition 2.1.8, with supp(f ϕ) {ξ R n : 1 (1 + a) 2 ξ 2(1 a)} and (F ϕ)(ξ) = 1 if 1 a ξ 1 + a, for some a > 0. Consider e 1 = (1, 0,..., 0) R n and let (a j ) j N C be satisfy a j c r 2 jr for some r > 0. Consider ζ S(R n ) and for all j N define ζ j := ζ( 2 j e 1 ). For all j 1 N we define (convergence in S(R n ), in particular pointwise convergence) the function Υ j1 := j 1 a jζ j F ϕ j1. Then, for all positive number b, it holds in S(R n ) the convergence Proof of Lemma lim j 1 2j 1b (Υ j1 a j1 ζ j1 ) = 0. (i) Let j 1 N fixed. First we prove the convergence of the sum Υ j1 in S(R n ). We notice that supp(f ϕ j1 ) B 2 j 1 +1(0) and that D β (F ϕ j1 ) S(R n ), β (N 0 ) n. Let j 2, j N with j j 2 j Then we have ξ 2 j e 1 2 j ξ 2 j 2 j j 1, ξ B 2 j 1 +1(0). For all multi-indexes β (N 0 ) n we have sup R n ξ 1+r D β ζ(ξ) c r,β and then (pointwise convergence) j j 2 sup B2 j 1 +1(0) 2jr D β ζ j (ξ) j j 2 sup B2 j 1 +1(0) 2 jr c r,β ξ 2 j e 1 1+r j j r 2 jr c r,β 2 j(1+r) = c r,β 2 j 2. Hence, for all β (N 0 ) n we obtain the inequalities (pointwise convergence) sup R n aj j j 2 D β (ζ j F ϕ j1 )(ξ) β β j j 2 c r,β sup B2 j 1 +1(0) 2jr D β ζ j (ξ) β β c r,β = c r,β. 2 j 2 2 j 2 Because of this uniform convergence we can commute the sum and derivative op- 99

109 5.5 Proofs of Section 4.2 and Theorem Longer Proofs erator in the series j 1 a jd β (ζ j F ϕ j1 ), for all β (N 0 ) n. Υ j1 C 0 (R n ) S(R n ). In particular we have Furthermore, as sup R n sup β l ξ m j j 2 a j D β (ζ j F ϕ j1 )(ξ) c r,l,m 2 j 2 holds for all l N 0 and all m N 0, then the series Υ j1 converges in S(R n ). (ii) Let j 1 N and consider k N 0, l N 0 and m N 0. We have the estimation sup β l ( ξ k D β ζ(ξ) ) c k,l, ξ R n, and the equality F ϕ j1 = (F ϕ)(2 j 1 ) gives sup j1 1 sup β l D β (F ϕ j1 )(ξ) sup β l D β (F ϕ)(ξ) c l, ξ R n. On the other hand, having in mind the decomposition ( ) Υ j1 a j1 ζ j1 = j 1 j j a 1 jζ j F ϕ j1 + a j1 ζ j1 (F ϕ j1 1), and taking into account the considerations in (i), we observe that for all ξ R n the quantity sup β l 2 j 1b ξ m D β (Υ j1 a j1 ζ j1 )(ξ) can be estimated from above by c r,l 2 j 1b ξ m j 1 j j 1 2 jr sup α l D α ζ j (ξ) sup D β (F ϕ j1 )(ξ) + β l + c r,l 2 j1b 2 j1r ξ m sup D α ζ j1 (ξ) sup D β (F ϕ j1 1)(ξ). (6) α l β l The first term of (6) equals 0 if ξ 2 j 1+2. Hence, by choosing k > r we can estimate this term from above by c r,l,m 2 j 1b 2 j 1m ( j1 1 j=1 c k,l 2 jr ( a 2 2j 1 ) k c l + j j 1 +1 ) c k,l 2jr c (2a2 j ) k l c a,r,k,l,m 2 j 1(b+m+r k) + c a,r,k,l,m 2j 1(b+m (k r)). c a,r,k,l,m 2j 1(r+b+m k). In the second term of (6), the second supremum equals 0 if ξ 2 j 1 e 1 2 j 1 a. On the other hand, with ξ 2 j 1 e 1 2 j 1 a, by Lemma we have ξ 2 j 1 e 1 c a ξ, where c a is a strictly positive constant independent of j 1 and ξ. Hence this second term can be estimated from above by c k,l c r,l 2 j1b 2 j1r ξ m (max{2 j 1 a, ca ξ }) c l. k This term can be evaluated from above by c a,r,k,l,m 2 j 1b 2 j 1r 2 j 1m 2 j 1k = c a,r,k,l,m 2 j 1(r+b+m k) if c a ξ 2 j 1 a, and by c a,r,k,l,m 2 j 1b 2 j 1r ξ m k c a,r,k,l,m 2j 1(r+b+m k) if c a ξ 2 j 1 a and 100

110 5.5 Proofs of Section 4.2 and Theorem Longer Proofs k m. Then we obtain finally sup R n sup(2 j1b ξ m D β (Υ j1 a j1 ζ j1 )(ξ) ) c a,r,k,l,m 2 j1(r+b+m k). β l By choosing k > r + b + m we have that this term tends to 0 when j 1, so the Proof is complete. Observation: We observe that if in Lemma we impose the additional hypothesis supp(f ϕ) {ξ R n : 1 a ξ 1 + a}, then we have Υ j1 a j1 ζ j1 j 1 N. In that case the proof of the Lemma would be immediate. = 0 for all Lemma Let 0 < p and ψ S(R n )\{0}, and for all j N consider θ j R and define W j (x) := ψ(x) cos(2 j x 1 + θ j ), x R n. Then we have W j Lp (R n ) j 1. Proof of Lemma Let j 0 N sufficiently large and j j 0. Then the period of cos(2 j x 1 + θ j ) relatively to x 1 is no greater than 2π2 j 0. Hence by an adequate translation, of k2π2 j 0 for some k Z relatively to x 1 -direction, we may assume that ψ(0) 0. By the way, we may assume ψ real and satisfying ψ(0) = 2. Then j 1 N such that ψ(x) 1, x [0, 2π2 j 1 ] n. Consider j j 1. Thus W j Lp (R n ) ( [0,2π2 j 1] n cos(2 j t 1 + θ j ) p dt) 1 p = (2π2 j 1 ) n 1 p ( [0,2π2 j 1] cos(2j t 1 + θ j ) p dt 1 ) 1 p = cj1. Because W j Lp (R n ) ψ Lp (R n ) for all j N, the Proof is complete. Proof of Theorem Consider temporarily ζ := F ψ, and for all j N define W j (x) := ψ(x) cos(2 j x 1 + ( θ j ), x R n. Then we have F W j = 1 2 e iθ j ζ( 2 j e 1 ) + e iθ j ζ( + 2 j e 1 ) ), j N. Let b > 0 and j 1 N, and consider ϕ as in Lemma We notice first that ϕ j1 S(R n ), W j S(R n ), and ψw = j 1 a jw j (pointwise convergence and also convergence in S (R) with the strong topology to the same function). By a direct generalization of Lemma 5.5.2, the series j 1 a j(f W j )(F ϕ j1 ) converges 101

111 5.5 Proofs of Section 4.2 and Theorem Longer Proofs in S(R n ), and it holds in S(R n ) the convergence (( lim j1 2 j 1b ) ) j 1 a j(f W j )(F ϕ j1 ) a j1 F W j1 = 0. By applying F 1 (the continuity of F 1 on S(R n ) considered both for the limit and for the sum) and taking into account on S(R n ) the property F (Φ Ψ) = (2π) n 2 (F Φ)(F Ψ), then we obtain in S(R n ) the convergence (( lim j1 2 j 1b ) ) j 1 a jw j ϕ j1 a j1 (2π) n2 W j1 = 0. Applying to the sum the continuity of the convolution on S (R n ), we obtain in S(R n ) that lim j 1 2j 1b (ϕ j1 (ψw ) a j1 (2π) n 2 Wj1 ) = 0. By the continuity of Lp (R n ) : S(R n ) R we obtain lim ( ) j 1 2j 1b ϕ j1 (ψw ) a j1 (2π) n 2 Wj1 Lp(R n ) = 0. On the other hand, by setting temporarily α j1 := ϕ j1 (ψw ) and β j1 := a j1 (2π) n 2 W j1, and taking into account that Lp (R n ) and lq ( j1 1 ( 2 j 1 s β j1 Lp(R n )) q ) 1 q ( ( ) ) 1 c j j 1 s q q α j1 Lp(R n ) + c ( j 1 1 ( 2 j 1 s β j1 α j1 Lp (R n ) (7) are quasi-norms, it comes out that ) ) 1 q q, and, of course, it holds also the analogous estimation obtained by interchanging the roles of α j1 and β j1. Hence, by taking b > s we prove the double implication ( 2 j 1 s q α j1 Lp (R )) n < ( 2 j 1 ) s q β j1 Lp (R n ) <. (8) j 1 1 j 1 1 Thus, by Lemma we proved nothing else than the equivalence ψw Bp,q(R s n ) j 1 1 (2j 1s a j1 ) q < (standard modification if q = ), so the Proof is complete. Definition Let ρ > 1. We generalize several definitions of Section 2 by replacing the numbers N and 2 by the number ρ, and adding the index ρ whenever convenient. (a) In Definition (or, alternatively, in Lemmas 5.2.2/5.2.5 with Definition 5.2.1), by replacing the number 2 by ρ we define the functions ϕ ρ and ϕ ρ j for j N 0, and 102

112 5.5 Proofs of Section 4.2 and Theorem Longer Proofs the spaces B ρ,s p,q(r n ). In the same way we define the corresponding classes B ρ,s p (R n ), Bp ρ,s+ (R n ) and Bp ρ,s (R n ). Remark: ([31], p. 46) The linear spaces B ρ,s p,q(r n ) are independent (equivalent quasinorms) of ρ. (b) Like in Definition we define I ρ := n k=1 [(j k 1)ρ ν, j k ρ ν ] where j k {1,..., [ρ ν ]} and ν N 0. If in this definition of I ρ we put j k Z instead, then we obtain I ρ as a corresponding generalization of Definition (c) As in Definitions and 2.2.4, but replacing N by ρ, we define Osc ρ (ν, f) := I ρ =ρ νn osc Iρ (f), and the spaces V ρ,α (T ) and V ρ,α (R n ) by the respective semi-norms f V ρ,α (T ) := sup ν 0 ρ ν(α n) Osc ρ (ν, f) and f V ρ,α (R n ) := sup ν 0 ρ ν(α n) I ρ =ρ νn osc I ρ (f). We define also the corresponding classes V ρ,α (R n ), V ρ,α+ (R n ) and V ρ,α (R n ). Remark: Although not proved here, we have the counterpart for remark (b) of Definition 2.2.4, i.e., the linear spaces V ρ,α (T ) and V ρ,α (R n ) are independent (equivalent semi-norms) of ρ. (d) We generalize also Definition by defining the classes V ρ,α (R n ), V ρ,α (R n ), V ρ,α+ (R n ) and V ρ,α (R n ), and similarly B ρ,s p, (R n ), B ρ,s p (R n ), B ρ,s+ p (R n ) and B ρ,s p (R n ). Remark: We have the counterpart for remark (b) of Definition 2.3.1, i.e., the classes V ρ,α (T ) and V ρ,α (R n ) are independent (equivalent pseudo-norms) of ρ. Nevertheless, the classes B ρ,s p, (R n ) are not independent of ρ. In fact, by the equivalence of Remark (d), applied to W s (x 1,..., x n ) = j 1 2 (2j)s cos(2 (2j) x 1 ), it follows that ψw s B s p, (R n ) = B 2,s p, (R n ) holds for any s R, in particular for large positive s. Nevertheless, the generalization of this Remark (d), according to Theorem 5.5.4, applied to ρ = 4 and W s (x 1,..., x n ) = j 1 4 js cos(4 j x 1 ), gives also ψw s B 4,s p, (R n ) if and only if s s. (e) Finally, and similarly to Definition 2.4.2, if f : A R n R is a function such that Γ(f) is a non-empty and bounded subset of R n+1, we define the upper box dimen- log M sion dim B Γ(f) := lim ρ(ν,f) ν. In a corresponding way we define also the upper log ρ ν and lower box dimensions dim B F and dim B F for a general non-empty set F R n. Remark: These definitions for upper and lower box dimensions coincide with the given 103

113 5.5 Proofs of Section 4.2 and Theorem Longer Proofs ones in Definition In fact, they are independent (same value) of ρ, see e.g. [14], p. 41. Because of this, we omit the index ρ. Theorem Let ρ > 1. In the same way of Definition we generalize several results by replacing the numbers N and 2 by the number ρ. Hence we generalize the remarks of Definitions 2.1.8, and We generalize also Lemma 5.1.1, Remark 5.1.2, Theorem 3.2.1, Remark 3.2.2, Theorem (trivially); Lemmas 5.2.2/5.2.5 with Definition 5.2.1, Lemmas and 6.3.3/5.2.3/5.2.4, Theorem 3.3.1, Corollary 3.3.2, Remark 3.3.3, Theorem 3.3.4, Corollary 3.3.5, Remark 3.3.6; Lemmas 5.5.2/5.5.3, Theorem 4.2.1, Remark Proof of Theorem The generalization of these results follows with the same Proofs, by replacing the numbers N and 2 by the number ρ, whenever it does make sense according to Definition (Of course we do not change some constants, as for example 2π and (2π) n 2.) We need only in some cases take the integer part [ ] of a positive expression and use the equivalence [K] K K for all real K 1. Proof of Theorem (i) Let φ : R R with φ C 0 (R) \ {0}, consider the sequence s = (s j ) j N, and define W s : R R by W s (x) := j 1 2 js j cos(2 j x + θ j ). Then Theorem gives φw s B s p (R) for 0 < p. (Modification for s =.) For the function f = φw s we apply Corollary with in place of p and apply also Corollary with 1 in place of p, obtaining dim B Γ(φW s ) = 2 min{1, s}. If in place of φ we put φ 1 such that supp φ 1 [0, 1] and φ 2 such that φ 2 (x) = 1 if x [0, 1], then we get dim B Γ(φ 1 W s ) dim B Γ(W s [0,1] ) dim B Γ(φ 2 W s ), where the first inequality is justified by Theorem (a). Therefore we have dim B Γ(W s [0,1] ) = 2 min{1, s}. On the other hand, the equivalence of Remark (d) gives φw s B s p (R) for 0 < p. (Modification for s =.) For the function f = φw s we apply also Remark (b) with 1 in place of p, getting dim B Γ(φW s ) 2 min{1, s}. By choosing φ 104

114 5.5 Proofs of Section 4.2 and Theorem Longer Proofs such that supp φ [0, 1], the Theorem (b) gives dim B Γ(W s [0,1] ) dim B Γ(φW s ), which gives the desired inequality. (ii) Consider ρ > 1. In the sense of Theorem with Definition 5.5.1, we can generalize part (i), including the references, by replacing the number 2 by the number ρ. (Of course we do not change the constant 2 in the expression 2 min{1, }.) By taking into account the independence of ρ given in remark of Definition (e), the Proof is now complete. We present the following elementary Lemma 5.5.5, which will be useful below in the preliminary part (i) of the Proof of Theorem Lemma Let (a j ) j N, (b j ) j N be two increasing sequences of strictly positive numbers, satisfying lim j b j /a j = 1. Then, it holds lim j j j =1 b j / j j =1 a j = 1. In the Proof of the Theorem we will employ as far as possible geometric arguments. This is not surprising since the very notion of Hausdorff dimension, as well as the related concept of mass distribution, are precisely based on geometric formulations. Though it is not difficult to develop all implicit calculations, doing this we would loose a good deal of clarity in the presentation, without any gain of objectiveness. Proof of Theorem (i) Let us start by defining a Cantor-like set K [0, 1], in order to remove short intervals around local maximums and minimums of W s, as follows. Let us denote by K 0 and K 0 the two subintervals of monotony considered for g in the Theorem. Furthermore, in the following we assume, without any loss of generality, that the function g has period 1, and that K 0 and K 0 are similar intervals, i.e. they have the same length. Let j j(ρ, γ, s) := j(g, K 0, K 0, ρ, γ, s) be a large integer, and for each fixed j define the set K j of all x [0, 1] such that ρ γj x + θ j K 0 K 0 + k for some k = k x Z. Actually, we complete the definition of K j by removing from it all those x for which k x is maximal or minimal over the all taken k (j fixed). Thus, K j is an union of similar intervals in which W j (x) := g(ρ γj x + θ j ) satisfies, with alternated signs of monotony on consecutive intervals, the growth properties assumed for g on K 0 and K

115 5.5 Proofs of Section 4.2 and Theorem Longer Proofs So, the same growth behavior is satisfied by W j (x) := j s j =1 ρ γj g(ρ γj x + θ j ) on K j := j j =j(ρ,γ,s) K j. Actually, we can write K j also as an union of intervals I j and, as before, we must complete the definition of K j by removing from it all those intervals I j which have (due to a truncation in the intersection above) a strictly smaller length than the normal ones. In this way, we arrive to a Cantor-like set K := j j(ρ,γ,s) K j, which in fact is a h-set and satisfies dim H K = 1. In order to prove these two facts about K, we will deal with the mass distribution usually defined on Cantor sets, see e.g. [14], pp We start by writing K j(ρ,γ,s) as an union of similar intervals I j(ρ,γ,s), and by distributing uniformly the unit of mass of [0, 1] by these intervals I j(ρ,γ,s). For each j = j(ρ, γ, s), j(ρ, γ, s) + 1,... and for each fixed interval I j, we write I j K j+1 as an union of similar intervals I j+1 (removing the truncated ones) and then we distribute uniformly the mass of I j by these intervals I j+1. So we arrive to a mass distribution µ K supported on the Cantor-like set K. We define δ as the Lebesgue measure δ := λ(k 0 K 0 ), consequently the quantity 1 δ is the measure of the remaining part of a (unitary) period of g containing K 0 K 0. According to the construction above, the set K j contains (similar) intervals I j with length I j = (δ/2)ρ γj, whose mass can be calculated iteratively on j and is given by µ K (I j ) = (2[ρ γj(ρ,γ,s) ] + κ j(ρ,γ,s) ) 1 j j 1 (δ/2)ρ γ j =j(ρ,γ,s)+1 (2[ ] + κ j ) 1, ρ γj where κ j { 2, 1, 0, 1, 2} for each j. The sequence κ := (κ j ) j depends on the particular interval I j, and is related with the number of these intervals we remove in each iteration j, as described above. (The product Π above must be understood as the unity when j = j(ρ, γ, s) and, as usually, the notation [ ] stands for the integer part.) Observe that I j = (δ/2)ρ γj(ρ,γ,s) j ρ γj j =j(ρ,γ,s)+1 ρ γj 1 holds, and set a j := log ργ j ρ γj 1 and b j := log(2[ (δ/2)ργj ρ γj 1 ] + κ j ) for each j. Then, we can write µ K (I j ) = I j d j, where d j := log(µ K(I j )) log I j = c + j j =j(ρ,γ,s) b j c+ j. j =j(ρ,γ,s) a j As we have lim j b j /a j = 1, then the Lemma gives the equality lim j d j = 1. If we consider balls Ḃj R with Ḃj = r j j ρ γj, centered at any X K = supp µ K, then µ K (Ḃj) j r d j j, where lim j d j =

116 5.5 Proofs of Section 4.2 and Theorem Longer Proofs (We may assume 0 < d j < 1, as well as that d j does not depend on the particular Ḃj.) Considering now a small ball Ḃ r R with 0 < Ḃr = r < 1 and centered at any X K, we have r j+1 < r r j for some j = j r and, by taking into account the particular characteristics of the set K, we obtain the relation µ K (Ḃr) r r d j j r/r j = r r d j 1 j =: r dr. Because r d j = r r d j 1 r r d j 1 j = r d r, then it follows that d j d r < 1. Hence we have µ K (Ḃr) r r d r (so K is a h-set) with d r satisfying lim r 0 + d r = 1 and, as a direct consequence, by remark (b) of Definition we obtain the equality dim H K = 1. (In part (ii) we will not apply this equality, we will use only the relation lim j d j = 1.) (ii) Now, we will prove the equality of the Theorem. Looking to the estimation of Theorem concerning lower box dimension and to the comparison between lower box and Hausdorff dimensions in remark (a) of Definition 2.5.3, we realize that it is sufficient to prove that dim H Γ(W s K ) 2 Ξ(γ, s) holds. In order to prove this inequality let us start by defining, induced by the mass distribution µ K defined in part (i), and supported on the graph of W s K, the useful mass distribution µ 0 (U) := µ Ws K (U) := µ K ({x K : (x, W s (x)) U}), where U are Borel subsets of R 2. In the rest of the Proof, we will consider j j(ρ, γ, s) a sufficiently large integer, according to part (i). We will estimate from above the mass of small balls centered at any point P Γ(W s K ) = supp µ 0, considering first the balls B j R 2 with diameter B j = r j j ρ γj, see Figure 17 below. As we will see, these balls correspond asymptotically to the minimal mass. On the other hand, in part (iii) below we will be in an opposite situation when we calculate the mass µ 0 applied to appropriate balls B j R 2 with diameter, according to Figure 17, given by the relation B j = r j j ρ γjs cos α j 1 j ρ γjs /tan α j 1 j ρ γjs /ρ γj 1 (1 s). (Concerning these balls B j, see also Figure 13 and recall the calculations following it.) In fact, this class of balls with diameter r j j ρ γj 1 (1 s+γs) j r 1 s+γs 1 s+γs γ j 1 j r j, corresponds asymptotically to the maximal mass, which allow us to obtain a sharp estimation for dim H Γ(W s K ). We should observe that 1 < 1 s + γs < γ holds, where 107

117 5.5 Proofs of Section 4.2 and Theorem Longer Proofs the first inequality is obvious and the second follows from the relations 1 s + γs < γ(1 s) + γs = γ or alternatively, from the equality 1 s + γs = γ (γ 1)(1 s). Let l j(ρ, γ, s) be an integer number and assume additionally, without any loss of generality, that min(g) 0 and K 0 K 0 [0, 1]. A scale peak of the function W s K, or, shortly, a peak, denoted by Λ l = Λ (m) l, is by definition a subset of the graph of W s K, when restricted to K [(m θ l )ρ γl, (m + 1 θ l )ρ γl ] = K [I (m) l some integer m for which we have a non-empty intersection. Here, I (m) l I (m) l ] for and I (m) l two consecutive (similar) intervals I l of the set K l. The height of a peak Λ l, h Λl, obtained by elementary calculations, is given by l ρ γls. More precisely, we have the relation h Λl := max Λl min Λl l j l ±ρ γj s l ρ γls, where min Λl and max Λl stand for the minimum and maximum, respectively, of the vertical coordinate when we run over all points (x, W s K (x)) of the peak Λ l. Moreover, most of the peaks Λ l have two geometrical halves, coming from the two monotonic parts of W l (m) I, as can be graphically observed in Figure 17 where l I (m) l the two halves of a peak Λ j 1 of W s K are explicitly represented, as well as a large number of peaks Λ j. For a given j, standard calculations show that both graphs of W s K and W j K have geometrical peaks Λ j with height j ρ γjs, and that each peak Λ j 1 contains j ρ γj 1 /ρ γj many peaks Λ j. Let us now estimate the mass µ 0 = µ Ws K of balls B j 1 centered at any P Γ(W s K ), which are not explicitly represented in Figure 17 since they would appear too large. In order to do this, we define the angle α j 1, see Figure 17, by the relation W j 1 (y) W j 1 (x) j,x,y y x tan(α j 1 ) on any of the intervals I j 1 of K j 1. Therefore, tan(α j 1 ) j 1 s j j =1 ρ γj (±ρ ) γj j ρ γj 1 (1 s) holds on any interval I j 1, since (W j (y) W j (x)) j,x,y ±ρ γj (y x), 1 j j 1, holds on each I j 1. By considering a fixed one half Λ of a peak Λ j 1, and by taking into account the angle α j 1 on Λ, we can say, roughly speaking, that the vertical position of a peak Λ (m+1) j by an average amount are of Λ is obtained from the one of the preceding peak Λ (m) j, j := ρ γj tan(α j 1 ) j ρ γj 1 (1 s γ). 108

118 5.5 Proofs of Section 4.2 and Theorem Longer Proofs (In order to compare j with ρ γjs, observe that 1 s γ < γs, since 1 s < γ γs.) In more precise terms, by taking into account the value of tan(α j 1 ) coming from the partial sum W j 1 on the interval I j 1 analogously to that one coming from W j 1, as well as the height of every peak Λ j of (W s K W j 1 K ), estimated by j ρ γj s analogously to that one of a peak Λ j of W s K, then we have the relation min Λ k j C Λ j,k ρ γjs + k j, where the index k establishes a rank on the peaks Λ j of the half Λ, by starting from the lowest vertical position with k = 0. Here, C Λ stands for the minimum of the vertical coordinate, when we run over all points (x, W j 1 K (x)) of the corresponding fixed half Λ for the function W j 1 K, therefore C Λ is constant for Λ fixed. Because of that last relation, and since a ball B j 1 has height, or diameter, B j 1 = r j 1 j ρ γj 1, then the number N j of peaks Λ j that intersect a given ball B j 1 can be estimated from above by the quantity c ρ γ (ii1) Let γs 1. Hence N j c j 1 +hλj +ρ γj s ρ j γ j 1 +ρ γ j s j. ρ γj 1 (1 s γ) ρ γj 1 ρ γj 1 (1 s γ) = cργj 1 ( 2+s+γ). Therefore, we have µ 0 (B j 1 ) N j µ 0 (Λ j ) j N j ( ρ γj ) dj cρ γj 1 (2 s γ+γd j ) = cρ γj 1 (2 s ε j ), where ε j := γ(1 d j ), thus by (i) it holds lim j ε j = 0. So, µ 0 (B j 1 ) c B j 1 2 s ε j. (ii2) Let γs 1 and define J := max{j N 0 : ρ γj+j s ρ γj 1 } = max{j N 0 : γ j s γ 1 }. Because γs 1, then for l = j,..., j + J it holds ρ γj 1 +ρ γl s ρ γ l s ρ γl 1 (1 s γ) j,l = ρ γl 1 (1 s γ) ρ γl (1 s)(1 1 γ ). This quantity is, up to a strictly positive constant, an estimation from above of the number of peaks Λ l, which have height l ρ γls, that intersect a given ball B j 1 and are contained in any of the two halves of a fixed peak Λ l 1. So we have the inequality N j cρ γj (1 s)(1 1 γ ) and, on the other hand, for each l = j + 1,..., j + J + 1 the total numbers N l of peaks Λ l intersecting the fixed ball B j 1 can be recursively estimated: N j+1 cn j ρ γj+1 (1 s)(1 1 γ ) c ρ γj (1 s)(1 1 γ )(1+γ), 109

119 5.5 Proofs of Section 4.2 and Theorem Longer Proofs N j+2 cn j+1 ρ γj+2 (1 s)(1 1 γ ) c ρ γj (1 s)(1 1 γ )(1+γ+γ2), and so on, arriving to the inequality N j+j cρ γj (1 s)(1 1 γ ) J l=0 γl = cρ γj (1 s) γj+1 1 γ. Here, we must observe that the number J of iterations does not depend on j, so the product of constants coming from the previous steps is also independent of j. Finally, N j+j+1 cn j+j ρ γj 1 ρ γ(j+j+1) 1 (1 s γ) c ρ γj ((1 s)(γ J 1 γ ) 1 γ γj (1 s γ)) = c ρ γj ( 2+s γ +γ J+1 ), and then we obtain µ 0 (B j 1 ) N j+j+1 µ 0 (Λ j+j+1 ) j N j+j+1 ( ρ γj+j+1 ) dj+j+1 cρ γj ( 2 s γ γj+1 +γ J+1 d j+j+1) = cρ γj 1 (2 s ε j ), = cn j+jρ γj ( 1 γ +γj (1 s γ)) where ε j := γ J+2 (1 d j+j+1 ), thus by (i) we have lim j ε j = 0. Hence, it holds the inequality µ 0 (B j 1 ) c B j 1 2 s ε j. (ii3) By the two previous parts (ii1) and (ii2), we obtain for all j the estimation µ 0 (B j ) c B j 2 s ε j = c r 2 s ε j j, where lim j ε j = 0. Below, in the following part (iii), we will easily find an estimation for the mass µ 0 of the balls B j, see Figure 17, which have diameter B j = r j 1 s+γs γj j ρ γ 1 s+γs γ = rj. (iii) Taking into account the estimation obtained in part (ii) and the respective diameters of the balls B j and B j, then we get µ 0 (B j) r ( ) 2 c j supbj r j µ 0 (B j ) c r j 2 r s ε j j. Therefore, we obtain µ 0 (B j) c r 2 j r γ(s+ε j ) 1 s+γs j = c r 2 Ξ(γ,s) ε j j, where lim j ε j = 0. Consider now a small ball B r R 2 with 0 < B r = r < 1 and centered at any P Γ(W s K ) = supp µ 0. Suppose first r j r r j for some j = j r. Hence, we have µ 0 (B r ) c(r/r j ) 2 sup Bj µ 0 (B j ) c r 2 r s ε j j r c r 2 r Ξ(γ,s) ε j j c r 2 Ξ(γ,s) ε j. Suppose now r j r r j 1 for some j = j r. Then, by taking into account the particular characteristics of the graph of W s K and the respective diameters of the balls, 110

120 5.5 Proofs of Section 4.2 and Theorem Longer Proofs µ 0 (B r ) c(r/r j) sup B j µ 0 (B j) c r r 1 Ξ(γ,s) ε j j c r 2 Ξ(γ,s) ε j. In this way we obtain the estimation µ 0 (B r ) cr 2 Ξ(γ,s) ε r, where lim r 0 + ε r = 0. (This is the very opportunity to understand the Figure 16 (see at the end of Section 4.4), which illustrates the asymptotical qualitative behavior of the growth of the mass µ 0 on balls of radius r (0, 1).) Finally, by remark (b) of Definition we obtain dim H Γ(W s K ) 2 Ξ(γ, s) ε for all ε > 0, and by recalling the comments at the beginning of part (ii) the Proof is now complete. Proof of Remark (i) Part (a) follows by the Proof of Theorem with standard modifications, and part (b ) follows by a generalization of (b) and by part (iii) of that Proof with 1 in place of γ. Furthermore, we will prove the inequality of part (b) by applying the same techniques we used in that Proof. Likewise, we assume that g has period 1, min(g) 0, and K 0, K 0 [0, 1] are similar intervals. Additionally, we consider here ρ ρ(s) := ρ(g, K 0, K 0, s) sufficiently large. Hence, standard calculations show that there are geometrical scale peaks in the graph of the function W s K, where K [0, 1] is an appropriate Cantor-like set. If we consider balls Ḃj R with diameter Ḃj = r j j ρ j, centered at any X K = supp µ K, where µ K is the mass distribution associated to the set K, then for all j N, µ K (Ḃj) c r d j j, with d j := c +(j 1) log(2[(δ/2)ρ] 2) c +(j 1) log ρ, where the Lebesgue measure δ := λ(k 0 K 0 ) does not depend on ρ. Hence, by remark (b) of Definition we have dim H K lim j d j = log(2[(δ/2)ρ] 2) log ρ where c δ,ρ := log = 1 c δ,ρ log ρ, ρ 2[(δ/2)ρ] 2. (We may observe that c δ,ρ tends to log(1/δ) when ρ.) Let 0 < η < 1 and ρ(η) := ρ(δ, η) be such that c δ,ρ c δ,η := (1 + η) log( 1 ), ρ ρ(η). δ In what follows consider ρ ρ(s, η) := max{ρ(s), ρ(η)} and j j(ρ, η, δ) sufficiently large, so that the inequality d j 1 c δ,η log ρ holds. 111

121 5.5 Proofs of Section 4.2 and Theorem Longer Proofs We will estimate the mass µ 0 of balls B j 1 R 2 centered at any P Γ(W s K ), where µ 0 (U) := µ K ({x K : (x, W s (x)) U}) for Borel sets U R 2. In order to do this estimation, we start by observing that, for fixed j, the vertical position of a peak Λ (m+1) j is obtained from the position of the preceding peak Λ (m) j j ρ j tan α j 1 j ρ j ρ (j 1)(1 s) = ρ (1 s) ρ js. by an average amount Because every peak Λ j has height j ρ js, then the number N j of peaks Λ j intersecting a given ball B j 1, with height, or diameter, B j 1 = r j 1 j ρ (j 1) = cρ j, can be estimated from above by c ρ j +ρ js ρ (1 s) ρ js j 1, therefore we can write N j c. Consider J j := max{j N 0 : ρ (j+j )s ρ j }, thus we have j s (j + J j )s j. For l = j + 1,..., j + J j, the number N l ρ j +ρ ls of peaks that intersect a given ball B j 1 can be estimated recursively as N l c 0 N ρ (1 s) ρ ls l 1 c 0 ρ 1 s N l 1, therefore we obtain finally the inequality N j+jj c ( c 0 ρ 1 s) J j = c c J j 0 ρ J j(1 s). (We observe here that c 0 := c 0(g, K 0, K 0 ) and c 0 := 2c 0 depend only on g, K 0, K 0.) Unlike the Proof of the Theorem in part (ii2), here the number J = J j depends on j, so we take into account the inequalities 1 s s c J j 0 j c j s = (ρ j ) log cs log ρ j 1 J j 1 s j, and the relations s j B j 1 c s log ρ, where c s := c 1 s s 0 and c s := log c s = 1 s log c s 0 (we can choose c s > 1, so c s > 0). Hence, µ 0 (B j 1 ) N j+jj µ 0 (Λ j+jj ) cn j+jj ( ρ (j+j j ) ) 1 c δ,η log ρ ε j c c J j 0 ρ J js ρ j ρ (j+j j)( cδ,η log ρ +ε j) j c J j 0 ρ (j+j j)s ρ j(1 s) ρ j 1 s ( cδ,η log ρ +ε j) j B j 1 c s log ρ ρ j ρ j(1 s) ρ j c δ,η s log ρ ρ j ε j s j B j 1 2 s c s log ρ c δ,η s log ρ ε j s, where ε j := 1 c δ,η d log ρ j+j j satisfies lim j ε j = 0. So, we have the estimation µ 0 (B r ) c B r 2 s c s +c δ,η /s ε log ρ r for all 0 < r < 1, where lim r 0 + ε r = 0. By remark (b) of Definition we obtain the inequality dim H Γ(W s K ) 2 s c s+c δ,η /s log ρ for 112

122 5.5 Proofs of Section 4.2 and Theorem Longer Proofs all ρ ρ(s, η) = max{ρ(s), ρ(η)}, therefore we can complete the proof of part (b) by considering the quantity c g,s defined by c g,s := max{c s + c δ,η /s, (1 s) log ρ(s, η)}. (ii) Finally we prove part (c), a specialized case for which we can improve some estimations of the proof of part (b), where we obtained dim H Γ(W s K ) 2 s ( 1 s log c s η log(1/δ) ) / log ρ whenever ρ ρ(s, η) = max{ρ(s), ρ(η)}. Let ρ s ρ(s, δ) large, let j j(ρ, η, δ), consider 0 < δ < 1, K 0 := [0, δ/2], K 0 := [ δ/2, 0] and assume, without loss of generality, Λ(x) as the distance from x to the nearest integer. The vertical position of a peak Λ (m+1) j is obtained from the position of the preceding peak Λ (m) j by an (exact) amount of ρ j tan(α j ) := ρ j j 1 (1 s) j =1 ±ρj ρ j ρ (1 (j 1)(1 s) ) ( ) (1 s) j 1 ρ j = ρ (1 s) ρ js 1 1, where we assume ρ 1 s > ρ 1 s 1 2. As the inverse of the factor between brackets is χ ρ,s := ρ 1 s 2 inequality N j c ρ j +ρ js /2 ρ (1 s) ρ js estimation N l 2N l 1 c ρ j +ρ ls /2 ρ (1 s) ρ ls then we have the χ ρ,s c, and for l = j + 1,..., j + J j we have that the χ ρ,s = N l 1 2c ρ j +ρ ls ρ (1 s) ρ ls χ ρ,s holds, so we obtain N j+jj c ρ J j(1 s) χ J j ρ,s. This conclusion follows from formula (9) below, which is deduced in the following way: If a > 1 then k 0 (1 + a k ) = c k 0 e k k 0 ln(1+a k) c k 0 e c k 0 k k 0 a k <, and therefore j+j j l=j+1 c ρ j + ρ ls ρ (1 s) ρ ls c j+j j l=j+1 Finally, if ρ ρ(s, η, δ) is large (including with ρ 1 s χ ρ,s = ρ 1 s 2 ρ ls ρ (1 s) ρ ls = cρj j(1 s). (9) > 2), then by part (i) with in place of c 0 it follows that dim H Γ(W s ) 2 s c ρ,s,η,δ log ρ holds with c ρ,s,η,δ := 1 s log χ s ρ,s + 1+η log( 1 ), whenever η, δ (0, 1). The Proof can now be s δ completed by elementary calculations, taking ε = ε(ρ, s, η, δ) > 0. Observation: Consider W s as in Remark (b), and we may recall that the part (b ) is a direct generalization of (b). As we just have the estimation 2 s c g,s / ln ρ dim H Γ(W s ) 2 s as mentioned before, the second (last) inequality is a well-known one, then we realize that it is of interest to know how to minimize the number c g,s. Fixed g and ρ and s and η, we observe the following concerning c g,s : When δ increases 113

123 5.5 Proofs of Section 4.2 and Theorem Longer Proofs then the values of c δ,η and of ρ(η) decrease, however the values of ρ(s) and of c s may increase, so we have a trade-off when intending to minimize c g,s. Therefore, even knowing that dim H Γ(W s ) = 2 s holds in many cases see e.g. Theorem 4.2.6, in general we do not know the actual value of dim H Γ(W s ) up to an (apparently unavoidable) additive constant given by c g,s / ln ρ. W s (x) } j ρ γj 1 (1 s γ) Ball B j : diameter j ρ γj Ball B j: diameter 1 s+γs γj j ρ γ Angle α j 1 : tanα j 1 j ρ γj 1 (1 s) α j 1 Figure 17: Graph of the restriction W s K to a Cantor-like set, according to the Proof of Theorem 4.2.7, where W s (x) = j 1 ρ γjs g(ρ γj x + θ j ) x Proof of Theorem (i) Suppose ρ := (ρ j ) j N R + and (θ j ) j N R, and consider S a : R n C defined 114

124 5.5 Proofs of Section 4.2 and Theorem Longer Proofs by the series S a (x) := j 1 a j cos(ρ j x 1 + θ j ), for all a := (a j ) j N C such that S a converges in S (R n ) at least for the weak topology. Define in the analogous way the series S b, by setting b j := a j ( 1 + ρ 2 j ) σ 2 for all j N. Then, for σ R we notice that the identity I σ (S a ) = S b holds in S (R n ). This follows from the identity I σ ( aj e ±i(ρ jx 1 +θ j ) ) = b j e ±i(ρ jx 1 +θ j ), the well known formula cos(u) = eiu +e iu 2 if u R, and the continuity of I σ according to remark of Definition (ii) Let a := (a j ) j N R be a sequence such that a l 1, let S a be the series as in (i), and consider S (1) a as the corresponding series when n = 1. By remark (b) of Definition with elementary adaptations, applied here with R n 1 in place of E, we have the equality dim Γ(S a ) = (n 1) + dim Γ(S (1) a ), where dim stands for box or Hausdorff dimensions, and therefore we can assume n = 1 in the following part (iii), where we deal only with this kind of series. Moreover, in order to prove the Theorem, we may assume n d 1 < d 2 < n + 1 (recall that we assumed also d 1 > n in part (b)), since for d 2 = n + 1 it is sufficient, in part (iii), to replace s by 0 and introduce the factor 1/j 2 in the trigonometric Weierstrass-type series and, in part (iv), to use j 2 instead of [ j δ+1] as well as 1/ ln j in place of 1/j δ in the function Θ. This can be made by an elementary adaptation of the calculations that we developed for the dimensions. Furthermore, it is convenient to recall here that the Theorems and and comments following them also hold for the limiting case s = 1. (iii) We prove part (b) for n = 1. We will make use of the Theorem 4.2.7, by taking into account the Lemma (b,b ). Consider ρ > 1, γ > 1, 0 < s < 1 and let W s : R R be defined by W s (x) := j 1 ρ γjs cos(ρ γj x + θ j ). Then, by Theorem and comments following it, we have dim B Γ(W s ) = dim H Γ(W s ) = 2 Ξ(γ, s), where Ξ(γ, s) = γs 1 s+γs by (i) we get the identity is a strictly increasing function on γ and s. On the other hand, I t W s (x) = j 1 ρ γj (s+tϑ j ) cos(ρ γj x + θ j ), 115

125 5.5 Proofs of Section 4.2 and Theorem Longer Proofs where t > 0, and ϑ j := log(1+ρ2γj ) 1/2 log(ρ γj ) 4.2.7, comments following it, and Remark (a), it holds satisfies lim j ϑ j = 1. Therefore, by Theorem dim B Γ(I t W s ) = dim H Γ(I t W s ) = 2 Ξ(γ, s + t), whenever s+t 1. By choosing (γ, s) satisfying d 1 = 2 Ξ(γ, s+t) and d 2 = 2 Ξ(γ, s), we obtain the cases 0 < t < t sup and 0 < t t max of part (b), which are fulfilled by function f := W s. Here, we took into account the Lemma 3.6.1, specially parts (b,b ). In order to prove the case t = d 2 d 1 of both parts (a) and (b), all we need is to use in the previous argument the function W s as in Theorem instead, since in this case the equalities above hold with s in place of Ξ(γ, s), behaving as a representative for the limiting case γ = 1. We consider here this series W s defined on all R, and choose an appropriate sequence (θ j ) j N satisfying the equality of that Theorem applied both for s and s + t. We must take into account also the comments following the respective Proof, thus considering both Hausdorff and box dimensions in the more general case. (iv) We still have to prove the case 0 < t < d 2 d 1 of part (a). Consider t > 0, and let W s : R n R be defined by W s (x 1,..., x n ) := j 1 ρ js cos(ρ j x 1 + θ j ), where ρ > 1 and 0 < s < 1. On the basis of Theorem and Remark (b) we get dim B Γ(W s ) = n + 1 s, and also dim B Γ(I t W s ) = n + 1 (s + t), whenever s + t 1. Similarly to Remark (c), let Θ : R n R be as in Definition 3.4.1, but replacing j 2 by [ j δ+1], and 1 ln j by 1 j δ with δ > 0, and put 1 j δ Θ j in place of Θ j, and similarly for Θ j,l, so Θ := j 1 l ( 1 ψ (2 j ( m(j, l))) j δ (uniform convergence), ), and the [ j δ+1] n 1 different values of l corre- 1 s where m(j, l) :=, 2,l j δ [j δ+1 ],..., s n,l [j δ+1 ] spond to the values of (s 2,l,..., s n,l ) {1,..., [ j δ+1] } n 1. We assume ψ S(R n ) \ {0}, as well as that F ψ is real, has radial symmetry, and with support contained in some annulus {ξ R n : r 1 ξ r 2 }, where r 1, r 2 > 0 in particular, ψ is a continuous real function. For that modified function Θ, similarly to Remark (c) we obtain dim B Γ(Θ) = n δ. Define f := W s + Θ on R n, and choose (δ, s) satisfying d 1 = n + 1 (s + t) and 116

126 5.5 Proofs of Section 4.2 and Theorem Longer Proofs d 2 = n δ > n + 1 s. Because of the last inequality, by some standard adaptations we have also dim B Γ(f) = n δ = d 2, since the dominant oscillations of f are originated from Θ. On the other hand we have the equality dim B Γ(I t W s ) = d 1, therefore we have also dim B Γ(I t f) = d 1, since I t Θ is a continuous real function of sufficiently bounded oscillation, see part (v). Therefore, the continuous function f fulfills part (a) of the Theorem. (v) Finally we must prove the statement above about I t Θ, particularly the estimation I =2 νn osc I (I t Θ) c2 ν(n 1), for all ν N. (10) (In particular, by applying, with elementary adaptations, the first equivalence of the Theorem 3.2.1, with 1 in place of γ, we easily obtain the equality dim B Γ(I t Θ) = n.) We consider first n = 1, thus we have Θ = j 1 1 ψ j δ j ( 1/j δ ), where ψ j := ψ(2 j ) for all j N. We observe that the graphs of the Schwartz real functions F ψ j F (I t ψ j ) = (1+ 2 ) t 2 F ψ j have radial symmetry, therefore both ψ j and ψ j := I t ψ j are real functions for each j N. In particular, we can write the equalities sup ν 1 I =2 ν osc I ψ j ( 1/j δ ) = R ψ j (x 1/j δ ) dx = R ψ j (x) dx = sup ν 1 I =2 ν osc I ψ j. Moreover, we will prove that it holds R ψ j (x) dx c2 ju, j N, for some u > 0. As usually, the notation ( ) stands for the classic derivative. Well, for fixed 0 < ε < 1, R ψ j (x) dx = R ψ j (x) (1 + 2 j x 1+ε ) sup R ψ j (x) (1 + 2 j x 1+ε ) R dx 1+2 j x 1+ε dx 1+2 j x 1+ε c ε 2 j 1+ε supr ψ j (x) (1 + 2 j x + 2 j x 2 ). By taking into account some elementary properties of the Fourier transform, coming directly from the integral properties, we estimate now the last supremum from above by sup R ψ j (x) + 2 j sup R xψ j (x) + 2 j sup R x 2 ψ j (x) c R ξ(f ψ j)(ξ) dξ + c2 j R ( ξ(f ψ j )(ξ) ) dξ + c2 j R ( ξ(f ψ j )(ξ) ) dξ c R ξ(f ψ j)(ξ) dξ + c2 j R (F ψ j)(ξ) dξ + c2 j R ξ(f ψ j) (ξ) dξ and 117

127 5.5 Proofs of Section 4.2 and Theorem Longer Proofs + 2c2 j R (F ψ j) (ξ) dξ + c2 j R ξ(f ψ j) (ξ) dξ. By the relations F ψ j = 2 j (F ψ)(2 j ) and (F ψ j )(ξ) = (1 + ξ 2 ) t 2 (F ψ j )(ξ), ξ R, and because the support of F ψ j does not contain the origin, we get the inequalities sup R ( F ψ j ) (ξ) supr (F ψ j ) (ξ) (1 + ξ 2 ) t 2 + sup R (F ψ j )(ξ)tξ (1 + ξ 2 ) t 2 1 c2 2j 2 jt + c2 j 2 j 2 j(t+2) = 2c2 2j 2 jt. We obtain sup R F ψ j c2 j 2 jt, and by calculations as just above, we get also sup R ( F ψ j ) (ξ) c2 3j 2 jt. As we assumed that F ψ has a compact support, then R ψ j (x) dx c ε 2 j 1+ε (2 j 2 j 2 j 2 jt + 2 j 2 j 2 j 2 jt + 2 j 2 j 2 j 2 2j 2 jt + 2 j 2 j 2 2j 2 jt + 2 j 2 j 2 j 2 3j 2 jt ), so finally we obtain the estimation R ψ j (x) dx c ε 2 j 1+ε 2 j 2 jt = c u 2 ju, where u > 0 if ε is sufficiently small. On the other hand, ψ j vanishes at infinity because it is a Schwartz function, so the inequality sup ψ j R ψ j (x) dx holds, showing that the series j 1 ψ j( 1/j δ ) converges uniformly on the real line. Furthermore, we have I t ( ψj ( 1/j δ ) ) = ψ j ( 1/j δ ), and by applying the continuity of the lifting operator I t on S (R n ), see remark of Definition 2.1.5, we obtain I t Θ = j 1 1 ψ j δ j ( 1/j δ ) with uniform convergence. In particular, I t Θ is a continuous real function on the real line and it holds I =2 osc ν I (I t Θ) j 1 I =2 osc ν I ψ j ( 1/j δ ), therefore I =2 ν osc I (I t Θ) j 1 c u2 ju c for all ν N, i.e. the function I t Θ has a bounded variation. We only need extend appropriately this result to all n N. Actually, we can obtain the estimation sup ν N I =2 ν osc I ψ j (, x 2,..., x n ) c u 2 ju, for all x = (x 2,..., x n ) R n 1, for some u > 0, where c u does not depend on x. This can be made analogously to the calculations above, with ψ j x 1 in place of ψ j, getting I =2 osc νn I (I t Θ) 2 ν(n 1) j 1 c u2 ju c2 ν(n 1), for all ν N. Remark: (a) As we can see in the Proof, the Theorem is also true if we define the lifting operator by using the expression ξ σ instead of (1 + ξ 2 ) σ 2, see Definition More- 118

128 5.6 Proofs of Section Longer Proofs over, it remains true for any other continuous and positive expression ϕ σ ( ξ ) satisfying for σ < 0 the relations lim ξ ϕ σ ( ξ ) = 0 and lim ξ ϕ σ( ξ ) ξ σ 2 R +. (b) We can prove the Formula (10), in part (v) of the Proof above, in a shorter way, by applying known results, at least if we use ψ as in Definition and additionally with radial symmetry. In fact, despite that we must prove that I t Θ is real, calculations like those in part (iii) of the Proof of Theorem give Θ B n 1 (R n ), then by the Lemma we obtain I t Θ B (n+t) 1 (R n ). In particular, it is clear by [31], p. 131, that I t Θ is a continuous function, and by elementary embeddings we have I t Θ B n 1,1(R n ), so the Remark (c) gives I t Θ V 1 (R n ) and the Formula (10) follows. However, in the direct proof of this Formula (10) given in part (v) above, we can also appreciate the exponential decay when j growths to the infinity of the contribution for the oscillation of each term of the series I t Θ = j 1 1 ψ j δ j ( 1/j δ ). 5.6 Proofs of Section 4.4 Proof of Theorem (i) For each j N let W j := ρ js ζ(ρ j +θ j ) and if j is large consider I j [0, 1] as an interval of length I j = P ζ ρ j, where P ζ is the period of ζ. Let j 1, j 2 N such that j 1 < j < j 2. Then we have osc Ij (W j ) = S ζ ρ js, osc Ij (W j1 ) S ζ ρj 1(1 s) I j = P ζ S ζ ρj 1(1 s) ρ j and osc Ij (W j2 ) = S ζ ρ j 2s, where S ζ := sup ζ inf ζ and S ζ is a Lipschitz constant of ζ, according to Definition Hence, it follows that osc Ij (W s ) S ζ ρ js j 1 j 1 =1 P ζs ζ ρj 1(1 s) ρ j j 2 >j S ζρ j2s = = ρ (S js ζ P ζ S ζ j 1 ) j 1 =1 ρ (j j 1)(1 s) S ζ j 2 >j ρ (j 2 j)s. But the quantity between brackets can be estimated from below by S ζ P ζs ζ ρ 1 s 1 S ζ ρ s 1. Then, if ρ ρ(ζ, s) is large we get osc Ij (W s ) c ζ,s ρ js and consequently the relation osc Ij (W s ) j ρ js. So, the minimum number N ρ (j, W s ) of squares as in Definition (b) (with volume ρ 2j ) needed to cover Γ(W s ) satisfies N ρ (j, W s ) j ρ js ρ j 1 ρ j = ρ j(2 s). Hence, by comments given in Definition (e) we have dim B Γ(W s ) = 2 s. 119

129 5.6 Proofs of Section Longer Proofs (ii) Suppose now that Γ(W s ) is a d-set with respect to some mass distribution µ, according to Definition Then by (i) and the remark of that Definition we have d = 2 s. For j N large, let I j [0, 1] be an interval with I j = 3P ζ ρ j and write I j = I j I j I j as an union of three intervals of length P ζ ρ j. By the relation osc Ij (W s ) j ρ js of (i), we can cover (I j R) Γ(W s ) with j 3 ρ js = 3ρ j(1 s) balls centered at points P ρ j ( ) 2 s Γ(W s ) and with diameter ρ j Pζ Then µ(i j R) c3ρ j(1 s) ρ P j ζ j ρ j j I j = µ 0 (I j R). Also by the same relation of (i), there exist j ρ j(1 s) disjoint balls contained in I j R, centered at points P (I j R) Γ(W s ), and with diameter ( 2 s min{ ρ j, P 2 ζρ j }. Hence µ(i j R) cρ j(1 s) min{ ρ j, P 2 ζρ }) j j ρ j j I j = µ 0 (I j R). So, we have µ(i j R) Ij µ 0 (I j R) and then µ(u 1 R) U1 µ 0 (U 1 R) for all Borel sets U 1 [0, 1], therefore µ(u) U µ 0 (U) for all Borel sets U R 2. By Definition the relation µ 0 (B r ) r r 2 s holds for all balls B r with 0 < B r = r < 1 and centered at any P Γ(W s ) = supp µ 0, so the Proof is complete. Remark: Consider (ρ j ) j N R + with ρ j+1 ρ j ρ(ζ, s) large and let W s : [0, 1] R be defined by W s (x) := j 1 ρ s j ζ(ρ j x + θ j ). Then for this W s we have the implication of Theorem and the equality dim B Γ(W s ) = 2 s. (And if ρ j+1 ρ j have even dim B Γ(W s ) = 2 s.) additionally is bounded then we Proof of Theorem (i) We notice that if j N, k {1,..., 2H j }, and x 1, x 2 [(k 1)H j, kh j ], then by the construction of W V,H we have the inequality W V,H (x 1 ) W V,H (x 2 ) V j = (H j ) s V,H. Hence, by elementary calculations we obtain W V,H C s V,H (R) = B s V,H, (R). (This can be seen also by remark of Definition and by Theorem 2 of [36], p. 465.) (ii) Let us prove that Γ(W V,H [0,2] ) is a d-set, where d = 2 s V,H and W V,H is given by Definition Consider j N and fix k {1,..., 2H j }. Then we have inf [(k 1)H j,kh j ] W V,H = (b j,k 1)V j and sup [(k 1)H j,kh j ] W V,H = b j,k V j for some 120

130 5.6 Proofs of Section Longer Proofs b j,k {1,..., V j }. If we define m j N such that V (m j+1) < H j V m j, then for that fixed k {1,..., 2H j } we can just cover [(k 1)H j, kh j ] [(b j,k 1)V j, b j,k V j ], and therefore Γ ( ) W V,H [(k 1)H j,kh j ], with Ξj := V j = H j 0 juxtaposed rectangles R j with dimensions H j V m j. V m j j V j H j Observe that R j j H j for all such rectangles R j = R j,α, α = 1,..., Ξ j. On the other hand, and again for that fixed k {1,..., 2H j }, the division of [(k 1)H j, kh j ] [(b j,k 1)V j, b j,k V j ] into Ξ j rectangles originates a division of Γ ( ) W V,H [(k 1)H j,kh j ] into Ξ j geometrical parts. Each of these parts can in its turn be decomposed into H m j j 0 small parts, and therefore the overall Ξ j parts can be decomposed into Ξ j H m j j 0 = H m j j small parts, with the following property: given two of these H m j j small parts, either they differ only by a translation or by a reflection about the (vertical) Y -axis composed with a translation. So, their corresponding projections on the X-axis have equal Lebesgue measure L 0. (iii) Let µ 0 be defined by µ 0 (U) := λ ({x [0, 2] : (x, W V,H (x)) U}) for all Borel sets U R 2, where λ is the Lebesgue measure. We can see that µ 0 is a mass distribution on R 2 with Γ(W V,H [0,2] ) = supp µ 0. On the other hand, for all Ξ j rectangles R j,α = R j, α = 1,..., Ξ j, we have the same value for µ 0 (R j ), since µ 0 (R j ) = H m j j 0 L 0. So, we have the equalities Ξ j µ 0 (R j,α ) = µ 0 ( Ξ j α=1 R j,α ) = µ 0 ([(k 1)H j, kh j ] R) = H j and therefore the relations µ 0 (R j ) = H j Ξ j j H 2j V j j R j 2 (H j ) s V,H j R j 2 s V,H. Let B r be a ball centered at any P Γ(W V,H ) with 0 < B r = r < 1 and define j r := min{j N : R j r }. Now we consider all k {1,..., 2 2Hj }, so the ball B r covers at least one rectangle R jr = R jr,k,α and then µ 0 (B r ) µ 0 (R jr ) r R jr d, where d = 2 s V,H. Also we can cover B r Γ(W V,H [0,2] ) by θ rectangles R jr = R jr,k,α, where θ N does not depend on r. So we have µ 0 (B r ) θµ 0 (R jr ) r R jr d and therefore we obtain the relations µ 0 (B r ) r R jr d r r d. The Proof is complete. Remark: (a) Let µ 0 be as in part (iii) above, in the Proof of Theorem 4.4.6, and take into ac- 121

131 5.6 Proofs of Section Longer Proofs count a refinement of the arguments we used in part (ii) of that Proof. If := [(k 1)H j, kh j ] [a, b] is a rectangle contained in the full rectangle [(k 1)H j, kh j ] [(b j,k 1)V j, b j,k V j ], k {1,..., 2H j }, then the mass of, µ 0 ( ), is just proportional to the height h = (b a) of the rectangle, i.e., it holds the equality µ 0 ( ) = H j V j h. In order to prove this equality, consider u N satisfying u j. Then, by arguments used in part (ii) of that Proof, with u in place of m j, this proportionality is true when is of the form [(k 1)H j, kh j ] [(l 1)V u, lv u ], i.e. µ 0 ( ) = H j Ξ u = H j V j /V u all general as above. = H j V j V u = H j V j h ; consequently the result follows for (b) Let (V j ) j N N \ {1} and (k 0,j ) j N N be two any (not necessarily bounded) sequences, and for each j N consider H 0,j := 2k 0,j + 1 and H j := H 0,j V j. Suppose also that in Definition we use in each iteration j N the numbers V j and H 0,j, instead of V and H 0, respectively. Then, the equality (proportionality) in part (a) remains true if we replace V j and H j by j j =1 V j and j j =1 H j, respectively. Proof of Corollary (i) Consider two bounded sequences (V j ) j N N \ {1} and (k 0,j ) j N N, in such way that j j =1 V j j j j =1 Hs j, where H 0,j := 2k 0,j + 1 and H j := H 0,j V j for all j N. In order to get this, it is sufficient that for each j N we choose V j and k 0,j as follows: If j j =1 V j j j =1 Hs j Let now W s := W (1) s we choose V j+1 H s j+1, otherwise we choose V j+1 H s j+1. : R R be as in Definition but by using in each iteration j N the numbers V j and H 0,j instead of V and H 0, respectively. Then, for n = 1 the Corollary follows by the same arguments of the Proof of Theorem 4.4.6, based in the measure µ (1) 0 := µ 0, but with j j =1 V j respectively. (ii) For n N \ {1}, consider the function W s W (n) and j j =1 H 0,j in place of V j and H j 0, := W (n) s : R n R defined by s (x) := W s (1) (x 1 ) n i=2 Λ(x i) for all x = (x 1,..., x n ) R n, where W s (1) is the function considered in (i) satisfying the Corollary for n = 1, and Λ(t) := t + 1 if 1 t 0, Λ(t) := 1 if 0 t 1, Λ(t) := 2 t if 1 t 2, and Λ(t) := 0 otherwise. Let us 122

132 5.7 Proofs of Section Longer Proofs prove that Γ(W (n) s [0,1] n) is a d-set with d = n + 1 s. For this consider the measure ) for all Borel sets µ (n) 0 (U) defined by µ (n) 0 (U) := λ (n) ( {x [0, 1] n : (x, W (n) s (x)) U} U R n+1, where λ (n) ( ) is the Lebesgue measure on R n. Let Q (n) r R n+1 be a cube with sides parallel to the axes and centered at any P (n) := (X 1,..., X n, X n+1 ) Γ(W s (n) [0,1] n) = supp µ (n) 0. If r (0, 1) is the side length of Q (n) r and S (n) := {x [0, 1] n : (x, W s (n) (x)) Q (n) r }, then we have the identity S (n) = S (1) [X 2, X 2 ]... [X n, X n], where each [X i, X i ], i = 2,..., n, is an interval of the form [X i, X i ] := [max{0, X i r }, min{1, X 2 i + r }]. (We consider here 2 S (1) := {x 1 [0, 1] : (x 1, W s (1) (x 1 )) Q (1) r }, and Q (1) r R 2 as the square originated by the projection of Q (n) r on the plane X 1 X n+1.) Therefore we have µ (n) 0 (Q r (n) ) = λ (n) (S (n) ) = λ (1) (S (1) ) n i=2 λ(1) ([X i, X i ]), where λ (1) is the linear Lebesgue measure. On the other hand, part (i) and elementary calculations give λ (1) (S (1) ) = µ (1) 0 (Q (1) r ) r r 2 s. Hence µ (n) ) r r 2 s r n 1 = r n+1 s. By elementary calculations we have also µ (n) centered at any P (n) Γ(W (n) s 0 (B r (n) 0 (Q (n) r ) r r n+1 s, where B (n) r R n+1 are balls [0,1] n) with 0 < B (n) = r < 1. The Proof is complete. r 5.7 Proofs of Section 4.5 Proof of Theorem (i) Suppose first n = 1, and consider c 1 δ 2 ε h(δr)/h(r) c 2 δ 1+ε for all 0 < δ, r 1, for some ε > 0. (i1) Let f (1) := W h : R R be constructed as in Definition 4.4.2, but with V j N \ {1}, H 0,j := 2k 0,j + 1 (k 0,j N) and H j := H 0,j V j, in each iteration j N, instead of fixed V, H 0 = 2k and H = H 0 V, respectively. We claim that there are such sequences (V j ) j N N \ {1} and (k 0,j ) j N N, as two bounded sequences satisfying r 2 j j j =1 V j j h(r j ), where r j := j j =1 H 1 j, j N. In order to prove this, choose V N \ {1} such that V ε 3 1 ε c 2, and k 0 N such that H 0 ε 2 1 ε c 1 1 ( H 0 := 2 k 0 + 1). Furthermore, in each iteration take V j+1 := V and H 0,j+1 := 3 (thus c 2 H 1 ε j+1 V j+1) if r 2 j j j =1 V j h(r j), and take V j+1 := 2 and 123

133 5.7 Proofs of Section Longer Proofs H 0,j+1 := H 0 (so c 1 H ε j+1 V j+1 ) if r 2 j j j =1 V j h(r j). In the first one of the two situations we have rj+1 2 j+1 j =1 V j h(r j+1 ) V h(r j ) r 2 j j j =1 V j h(3 1 V 1 r j ) h(r j ) V c 1 1 (3 V ) 2 1 = c. But we obtain also, by using the hypothesis we assumed on V, the relations j+1 h(r j+1 )/h(r j ) c 2 (r j+1 /r j ) 1+ε = c 2 H 2 j+1 H1 ε j+1 H 2 j+1 V j+1 = r2 j+1 j+1 and therefore r2 j+1 j =1 V j h(r j+1 ) j r2 j j =1 V j h(r j ) j+1 j =1 V j h(r j+1 ) In the second situation we have r2 j+1 holds. (2 H 0 ) 2 r2 j j =1 V j rj 2 j, j =1 V j j j =1 V j h(r j ) (2 H 0 ) 2 1 = c. By taking into account the hypothesis we assumed on H 0, we obtain also the relations j+1 h(r j+1 )/h(r j ) c 1 (r j+1 /r j ) 2 ε = c 1 H 2 j+1 Hε j+1 H 2 j+1 V j+1 = r2 j+1 j =1 V j rj 2 j, j =1 V j j+1 and therefore r2 j+1 j =1 V j h(r j+1 ) j r2 j j =1 V j h(r j ) holds. In this way we get r2 j j j =1 V j h(r j ) j 1, and therefore the relation r 2 j j j =1 V j j h(r j ). On the other hand, by the arguments we used in the Proof of Theorem 4.4.6, but with j j =1 V j, j j =1 H 0,j, j j =1 H j in place of V j, H0, j H j, respectively, we have that the relation µ 0 (B j ) j r 2 j j j =1 V j holds for all balls B j with B j = r j and centered at any P supp µ 0 = Γ(W h [0,1] ). Here, µ 0 := µ (1) 0 is a mass distribution defined by µ (1) 0 (U) := λ (1) ({x [0, 1] : (x, W h (x)) U}) for all Borel sets U R 2, where λ (1) ( ) represents the Lebesgue linear measure. Consequently, we have µ 0 (B j ) j h(r j ). As the sequence ( r j r j+1 ) j N is bounded, then we obtain µ 0 (B r ) r h(r), for all balls B r R 2 with 0 < B r = r 1 and centered at any P supp µ 0 = Γ(W h [0,1] ). We proved that Γ(W h [0,1] ) is a h-set. (i2) We prove shortly and easily the regularity condition. Consider x, y R, with = y x x,y r j for some j = j N. Then, by construction see at the beginning of part (i1) we have W h (y) W h (x) j j =1 V 1 j j rj 2 /h(r j ) j 2 /h( ). (ii) Consider now the case n N\{1}, and define h (1) (r) := h(r)/r n 1, with 0 < r 1. (ii1) We have the equality h (1) (δr)/h (1) (r) = 1 δ n 1 h(δr)/h(r), and therefore we obtain c 1 δ 2 ε = c 1 δ n+1 ε δ n 1 h (1) (δr)/h (1) (r) c 2 δ n+ε δ n 1 = c 2 δ 1+ε, for all 0 < δ, r

134 5.7 Proofs of Section Longer Proofs Then, by part (i) there is a function f (1) := W h (1) : R R satisfying f (1) (x) f (1) (y) c 2 /h (1) ( ), and a mass distribution defined by µ (1) 0 (U) := λ (1) ({x [0, 1] : (x, f (1) (x)) U}), such that µ (1) 0 (B r ) r h (1) (r), for all balls B r R 2 with 0 < B r = r 1 and centered at any P supp µ (1) 0 = Γ(f (1) [0,1] ). Let Λ : R R be defined by Λ(t) = t + 1 if 1 t 0, Λ(t) = 1 if 0 t 1, Λ(t) = 2 t if 1 t 2, and Λ(t) = 0 otherwise. We will now to apply an argument similar to that one used in part (ii) of the Proof of Corollary Consider the function W h := f (n) : R n R, where f (n) (x) := f (1) (x 1 ) n i=2 Λ(x i), x = (x 1,..., x n ), and define the mass distribution µ (n) 0 (U) := λ (n) ({x [0, 1] n : (x, f (n) (x)) U}) for all Borel sets U R n+1, where λ (n) ( ) is the Lebesgue measure on R n. Let Q (n) r R n+1 be a cube with sides parallel to the axes and centered at any point (X 1,..., X n, X n+1 ) supp µ (n) 0 = Γ(f (n) [0,1] n). If r is the side length of Q (n) r, then we obtain µ (n) 0 (Q (n) r ) r r n 1 µ (1) 0 (Q (1) r ) r r n 1 h (1) (r) = h(r), where Q (1) r square originated by the projection of Q (n) r on the plane X 1 X n+1. R 2 is the In this way, we finally get µ (n) 0 (B r ) r h(r), for all balls B r R n+1 with 0 < B r = r 1 and centered at any P Γ(f (n) [0,1] n). We proved that Γ(W h [0,1] n) is a h-set. (ii2) Similarly to part (i2), we consider x, y R n with = y x x,y r j for some j = j N. We have h (1) (r) = h(r)/r n 1, as well as the relation W h (y) W h (x) j j =1 V 1 j + c y x j rj 2 /h (1) (r j ) = rj 2 r n 1 j /h(r j ) j n+1 /h( ), and therefore the Proof is now complete. Definition Let K R be a h 1 -set, and consider a mass distribution µ := µ K associated to K, see Definition Given a function f : [0, 2] R, then we consider T K f : K R defined by (T K f)(t) := f(τ(t)), where τ (µ) := τ : K [0, 2] is given by τ(t) := 2µ( ], t] ). have µ(r) = 1.) We call T (µ) K Remark: (We assumed here, without any loss of generality, that we := T K the domain operator, see Figures 18ab. We should observe the following: If U is a Borel subset of R, then τ(k U) is also a Borel set in R. This will be convenient below, when we define the mass distribution µ

135 5.7 Proofs of Section Longer Proofs Proof of Theorem (i) Given h 2 : (0, 1] R + continuous, and satisfying h 2 (0 + ) = 0 and c 1 δ 2 ε h 2 (δr)/h 2 (r) c 2 δ 1+ε for all 0 < δ, r 1, for some ε > 0. Then, the Proof of the Theorem gives us a function W h2 : R R satisfying W h2 (y) W h2 (x) c 2 /h 2 ( ), where = y x, such that the graph Γ(W h2 [0,2] ) is a h 2 -set, see Figure 18a. Now, consider the function T K W h2 : K R, according to Definition 5.7.1, see Figure 18b. (i1) Consider t, l K satisfying 0 < 1, where := l t. By elementary arguments we obtain, according to Definition 5.7.1, the inequality τ(l) τ(t) c h 1 ( ). As a consequence, we obtain (of course, we may assume sup (0,1] h 1 ( ) sufficiently small) T K W h2 (l) T K W h2 (t) = W h2 (τ(l)) W h2 (τ(t)) c h 2 1 ( )/h 2 (h 1 ( )). (We observe that for the interesting particular case of d-sets, h 1 (r) = r d 1, 0 < d 1 1, and h 2 (r) = r d 2, 1 < d 2 < 2, then T K W h2 (l) T K W h2 (t) c d 1(2 d 2 ).) (i2) In this part, we will use essentially the notation as in part (i) of the Proof of Theorem 4.5.1, here applied to the function W h2. For all Borel sets U R 2, consider µ 0 (U) := λ (1) ({x [0, 2] : (x, W h2 (x)) U}) with supp µ 0 = Γ(W h2 [0,2] ), and define µ 0 (U) := µ 0 ({(τ(t), w) : t K (t, w) U}), satisfying supp µ 0 = Γ(T K W h2 ). We notice here that by remark of Definition and some standard adaptations, this definition of the mass distribution µ 0 makes sense. Consider the mass distribution µ K associated to K, according to Definition Let us assume, without loss of generality, that h 1 is monotone and satisfies the inequality h 1 (r) µ K (Ḃr) (besides h 1 (r) r µ K (Ḃr)) for all balls Ḃr R with 0 < Ḃr = r 1 and centered at any X K = supp µ K. Furthermore, for all 0 < r 1 we define the integer j(r) := min{j N : 2r j h 1 ( r 2 )}, therefore the inequality µ K (Ḃ r 2 ) 2r j(r) holds. (Recall that r j = j j =1 H 1 j if j N see at the beginning of part (i1) of the Proof of Theorem ) In this way, the interval τ(k Ḃ r 2 ) contains (at least) an interval I k 1)r j(r), kr j(r) ] for some k {1,..., 2r 1 j(r) }, since I k = r j(r) := [(k and τ(k Ḃ r 2 ) 2r j(r). 126

136 5.7 Proofs of Section Longer Proofs Let P = (X P, Y P ) be a point belonging to the graph Γ(T K W h2 ), and suppose that B r R 2 is a ball with 0 < B r = r 1 and centered at P. Of course, the ball B r contains the square Q r := [X P side length X P r 2, X 2 P + r 2 ] [Y 2 P r 2, Y 2 P + r 2 ], which has 2 r 2. By particularizing the ball Ḃ 2 r = [X P r 2, X 2 P + r 2 ], centered at 2 K, we already know that τ(k Ḃ r 2 ) contains an interval I k and, as a consequence, the square Q r contains the rectangle := τ 1 (I k ) [Y P r 2, Y 2 P + r 2 ], where 2 τ 1 (I k ) R is the smallest interval that contains the set {t K : τ(t) I k }. Hence, µ 0 (B r ) µ 0 ( ) = µ 0 (I k [Y P r 2 2, Y P + r 2 2 ]) = µ 0(I k [a r, b r ]), where a r := max{y P r 2, inf 2 I k W h2 }, b r := min{y P + r 2, sup 2 I k W h2 }. On the other hand, by the remark that follows the Proof of Theorem 4.4.6, we obtain the relations (the remark just gives the first of the equalities, which in fact is a proportionality) µ 0 (I k [a r, b r ]) = j(r) j =1 H 1 j j(r) j =1 V 1 j (b r a r ) = br ar r j(r) rj(r) 2 j(r) j =1 V j b r a r b r r j(r) h 2 (r j(r) ) r a r r h h 1 (r) 2(h 1 (r)), see part (i1) of the Proof of Theorem concerning the second equality and the r following it the second (or last) r follows directly by the definition of j(r). At least for r > 0 sufficiently small (i.e. j(r) large) we have sup Ik W h2 inf Ik W h2 thus b r a r r 2, so we get the estimation µ 2 0(B r ) c r h 2(h 1 (r)) h 1 (r). r 2, 2 Concerning the converse inequality, we can cover B r Γ(T K W h2 ) with finite many rectangles (i.e. for finite many k) of the form := τ 1 (I k ) [Y P r, Y 2 P + r ], so we have 2 also the estimation µ 0 (B r ) c r h 2(h 1 (r)) h 1 (r). In this way, we obtain finally the relation r h µ 0 (B r ) 2 (h 1 (r)) r h 1 (r), for all balls B r R 2 with 0 < B r = r 1 and centered at any P Γ(T K W h2 ). Consequently, the graph Γ(T K W h2 ) is a ḧ-set, where ḧ(r) :=r h 2(h 1 (r)) h 1 (r). (In the case of d-sets, we have ḧ(r) = r d, where d := 1 + d 1 (d 2 1). We observe that d is strictly increasing on d 1 and on d 2. Furthermore, for fixed 1 < d 2 < 2, then d ranges over (1, d 2 ] for 0 < d 1 1, and for fixed 0 < d 1 1, then d ranges over (1, 1 + d 1 ) for 1 < d 2 < 2.) (ii) We assume, without any loss of generality, that h 1 is monotone and satisfies 127

137 5.7 Proofs of Section Longer Proofs sup (0,1] h 1 ( )= 1, and choose h 2 as in part (i), particularized by h 2 (h 1 (r)) := h 1(r)h(r) r. (In case of d-sets, where h(r) = r d, 1 < d < 1 + d 1, we choose d 2 such that d 1 d 2 = d 1 + d 1, in particular 1 < d 2 < 2 holds.) Hence, for this particular h 2, the hypotheses given at the beginning of part (i) are satisfied, and the identity ḧ = h holds for ḧ given as in part (i2). Therefore, by (i2) the graph of the function W h := T K W h2 is a h-set. Because we have h 2 1 (u)/h 2 (h 1 (u)) = u h 1 (u)/h(u) for all 0 < u 1 (u d 1(2 d 2 ) = u 1+d 1 d in the case of d-sets), then by part (i1) the Proof is complete. W(x) Figure 18a: This graph shows the general aspect of a h-set, W := W h, when constructed x as in Definition recall Figures 15abc and generalized according to the Proof of Theorem

138 6 Disc. Applic. Open problems K(x) Figure 18b: The graph W K := T K W originated from the one of Figure 18a, by applying K x K the domain operator T K, according to Definition (We don t see the details at a very small scale) 6 Discussion, applications and open problems 6.1 Summary of the Chapter In Section 6.2 we present results that, although out of the the main line of research, extend particular results of some previous Sections. We also state here the density Lemma and an application of it. We will use this Lemma in the next Section, in order to give a new proof (frequency approach) for Theorem In Section 6.3 we give new versions and new proofs for the embeddings of Section 3.3, and afterwards we compare frequency and wavelet approaches, and also the nonsmooth atoms approach, as far as concerns embeddings between oscillation and Besov spaces, estimations for fractal dimensions, or construction of d-sets as graphs. In Section 6.4, firstly we mention some practical contexts strongly related to fractals, and secondly we switch to practical applications which are inspired and based on the embeddings (and respective proofs) of Section 3.3. We consider the problem of the general signal detection, and then we present a numerical simulation for wave and chirp signals, for samples with dimension In Section 6.5 we list some open problems 129

139 6.2 Some complementary results 6 Disc. Applic. Open problems and conjectures that naturally arose from the obtained results in the main Sections 3 and 4, as well as conjectures for possible extensions or generalizations of that results. 6.2 Some complementary results In this Section we will state the density Lemma 6.2.3, which we will use in the Proof of Theorem given below in Section 6.3 where we use a frequency approach. Furthermore, in the following Lemma which states the result referred in [31], p. 23, Remark 3 we use that Lemma in order to easily extend Lemma from S Ω (R n ) to L Ω p (R n ), with 0 < p. (Recall Definition ) The following two Lemmas are the famous Paley-Wiener-Schwartz theorems. In part (i) of Lemma 6.2.2, if f is regular we can interpret supp both in the classic way or in the sense of the distributions. Lemma ([31], p. 13) The following two assertions are equivalent: (i) ϕ S(R n ) and supp F ϕ {ξ R n with ξ b}, (ii) ϕ extends to an entire analytic function of n complex variables still called ϕ and for any λ > 0 and any ε > 0 there exists a constant c λ,ε such that ϕ(z) c λ,ε (1 + x ) λ e (b+ε) y holds for all z = x + iy with x R n and y R n. Lemma ([31], p. 13) The following two assertions are equivalent: (i) f S (R n ) and supp F f {ξ R n with ξ b}, (ii) f extends to an entire analytic function of n complex variables still called f and for an appropriate real number λ and for any ε > 0 there exists a constant c ε such that f(z) c ε (1 + x ) λ e (b+ε) y holds for all z = x + iy with x R n and y R n. Lemma Let G, A R n be non-empty bounded sets and addictionally suppose that G is open. Hence, for all 0 < p <, it follows that S G (R n ) is dense in L G p (R n ) with the norm Lp (R n ) + sup A. Proof of Lemma Consider f L G p (R n ). Let ζ S(R n ) with ζ(0) = 1, ζ(x) 1, x R n and with 130

140 6.2 Some complementary results 6 Disc. Applic. Open problems supp(f ζ) compact. It is easy to find such a ζ, because a translation of ζ and a multiplication by a scalar k C \ {0} preserves supp(f ζ). For all k N let ψ k (x) := f(x)ζ(x/k), x R n. Then by Lemmas 6.2.1/6.2.2 we have ψ k S(R n ), k N. Because G is open, for k sufficiently large we have ψ k S G (R n ). Consider ε > 0. Define M ε > 0 with t M ε f(t) p dt 1 2 ( ε 2 )p, and define k ε N such that for all k k ε it holds ζ(x/k) 1 p ε p /2 1+ t Mε f(t) p dt for all x Rn with x M ε or x A. We observe by Lemma that f is bounded on A. Hence ψ k f Lp(R n ) ε and sup A ψ k f ε sup A f, k k ε. Lemma ([31], p. 19) Consider 0 < p, and let Ω R n be a compact set. For any h > 0 and k = (k 1,..., k n ) Z n, define Q h k := {(x 1,..., x n ) R n with hk i x i < h(k i + 1) for i = 1,..., n}. Then, there exist three strictly positive numbers h 0, c 1 and c 2 such that (standard modification if p = ) c 1 ( k Z n ϕ(x k ) p) 1 p h n p ϕ Lp(R n ) c 2 ( k Z n ϕ(x k ) p) 1 p holds for all h with 0 < h < h 0, all sets (x k ) k Z n with x k Q h k, and all ϕ SΩ (R n ). Lemma Consider 0 < p, and let G R n be a bounded open set containing Ω as in Lemma Let h 0, c 1 and c 2 be the (strictly positive) numbers referred in that Lemma with the same Ω. Consider 0 < h < h 0, let x = (x k ) k Z n with hk j x j < h(k j + 1), j = 1,..., n, and for all f L G p (R n ) define Ξ(f) := ( k Z f(x n k ) p ) 1/p. Then, for all f L G p with Ξ(f) < we have (standard modification for p = ) c 1 Ξ(f) h n/p f Lp(R n ) c 2 Ξ(f). Proof of Lemma Suppose that G is not empty and consider f L G p (R n ). Let ε > 0 and define Ξ ε (f) as the sum Ξ(f) restricted to k Z n with k 1/ε. By the density Lemma 6.2.3, ϕ ε S G (R n ) with max{c1, c2} Ξ ε (ϕ ε ) Ξ ε (f) < ε and h n/p ϕ ε Lp (R n ) f Lp (R n ) < ε. h n/p ϕ ε Lp(R n ) c 2 Ξ(ϕ ε ). So, by Lemma it holds c 1 Ξ(ϕ ε ) On the other hand, c 1 Ξ(f) h n/p f Lp(R n ) equals the sum 131

141 6.2 Some complementary results 6 Disc. Applic. Open problems (c 1 Ξ(f) c 1 Ξ ε (f)) + (c 1 Ξ ε (f) c 1 Ξ ε (ϕ ε )) + (c 1 Ξ ε (ϕ ε ) c 1 Ξ(ϕ ε )) +(c 1 Ξ(ϕ ε ) h n/p ϕ ε Lp (R n )) + (h n/p ϕ ε Lp (R n ) h n/p f Lp (R n )), which can then be estimated from above by (c 1 Ξ(f) c 1 Ξ ε (f))+ε+0+0+ε. Therefore, the inequality c 1 Ξ(f) h n/p f Lp (R n ) + (c 1 Ξ(f) c 1 Ξ ε (f)) + 2ε holds. Moreover, as we can see in its Proof, that functions ϕ ε of Lemma satisfy ϕ ε (t) f(t), t R n. Thus, we have additionally Ξ(ϕ ε ) Ξ ε (ϕ ε ) Ξ(f) Ξ ε (f). The quantity h n/p f Lp (R n ) c 2 Ξ ε (f) equals the sum (h n/p f Lp (R n ) h n/p ϕ ε Lp (R n )) + (h n/p ϕ ε Lp (R n ) c 2 Ξ(ϕ ε )) +(c 2 Ξ(ϕ ε ) c 2 Ξ ε (ϕ ε )) + (c 2 Ξ ε (ϕ ε ) c 2 Ξ ε (f)) + (c 2 Ξ ε (f) c 2 Ξ(f)), which can then be estimated from above by ε+0+(ξ(f) Ξ ε (f))+0+(ξ(f) Ξ ε (f)). Therefore, the inequality h n/p f Lp(R n ) c 2 Ξ ε (f) + ε + 2(Ξ(f) Ξ ε (f)) holds. Finally, by taking the limits when ε 0 +, we get the desired inequalities. Remark: As we can see, in the Proof of Lemma we essentially made use of Lemmas and Moreover, now after proving Lemma 6.2.5, we can condense parts (i)- (ii) of the (second) Proof of Theorem given in Section 6.3 below, by using this Lemma instead of Lemmas 6.2.3/6.2.4 in that Proof. More applications of Theorem 5.5.4, and another kind of generalizations Remark Besides Theorem 5.5.4, we take the oportunity to make here some other easy and direct generalizations, although they may be not very pertinent. (a) Of course, like in Theorem we can also make a generalization with ρ > 1 in place of N and 2, of Lemma 3.4.2, Theorem with Definition 3.4.1, Remark 5.3.3, Remark 3.4.3; Theorem with Definition 3.4.2; Theorem with Definition 3.4.3, Remark 5.3.4, Theorem 3.4.6; Theorem with Definition 4.3.1; Theorem and Remark 3.5.2; finally part (b) of the present Remark. (b) We can also make another kind of generalizations. First let us define the linear spaces V α (R n ) and B s p, (R n ), substituting lim by lim in Definition Hence if 132

142 6.2 Some complementary results 6 Disc. Applic. Open problems f : R n R is measurable we define f V α (R n ) := lim ν N ν(α n) I =N νn osc I (f) and if f S (R n ) we define f B s p, (R n ) := lim j 2 js ϕ j f Lp(R n ). In the same way we define also the corresponding classes V α (R n ), V α+ (R n ), V α (R n ), the linear space V α (T ), and the classes V α (T ), V α+ (T ), V α (T ). Similarly we define the classes B s p (R n ) := σ<s Bσ p, (R n ), B s+ p (R n ) := σ>s Bσ p, (R n ) and B s p (R n ) := B s p (R n ) \ B s+ p (R n ). Then, with these definitions, we have the upper counterpart of remark of Definition and the upper counterpart of Theorem and Remark for dim B, V γ (T ), V γ+ (T ), V γ (R n ), V γ+ (R n ). We have the upper counterpart of Remark (b), with V, B and dim B in place of the lower counterparts V, B and dim B. We have also a kind of upper counterpart for Remark (c), the embedding V γp (R n ) B γ p, (R n ) for 1 p < and γ R, and also the corresponding lower counterpart embedding. We have the upper counterpart for Theorem 4.2.1/Remark 4.2.2, i.e., the equivalence ψw B s p, (R n ) lim j 2 js a j <. Extension of Theorem and Corollary The Theorem 4.4.6, with Definition 4.4.2, was generalized by Corollary and Theorems 4.5.1/4.5.2, in order to construct graphs which are d-sets and h-sets, respectively. We dealt only with bounded sequences (H j ) j N in the respective Proofs, because in this case we obtain appropriate estimates for the mass of the corresponding graphs. In the following Theorem we give a generalization of that Corollary in a different direction, by dealing with such constructions of graphs when those sequences are unbounded. Although we don t have the guarantee that such graphs are d-sets, we show that if that sequence (H j ) j N does not grow excessively fast then the Hausdorff dimension of such graphs remains d = n + 1 s, where s falls between 0 and 1 and stands for the Hölder smoothness of the graph as a function on R n. Theorem Consider two (not necessarily bounded) sequences (V j ) j N N \ {1} and (k 0,j ) j N N. For all j N, denote H 0,j := 2k 0,j + 1, H j := H 0,j V j, V (j) := j j =1 V j, H 0(j) := j j =1 H 0,j, H(j) := H 0(j)V (j) = j j =1 H j, and s j := 133

143 6.2 Some complementary results 6 Disc. Applic. Open problems log V (j)/ log H(j) (so H(j) s j = V (j)). Assume that it holds 0 < s := lim j s j < 1 and also s j s (i.e. H(j) s V (j)). Furthermore, as a growth condition suppose additionally that lim j H j H(j) δ = 0 holds for all δ > 0. Let now W := W (1) : R R be as in Definition but by using in each iteration j N the numbers V j and H 0,j instead of V and H 0, respectively. For n N \ {1}, consider the function W := W (n) : R n R defined by W (n) (x) := W (1) (x 1 ) n i=2 Λ(x i) for all x = (x 1,..., x n ) R n, where Λ(t) := t + 1 if 1 t 0, Λ(t) := 1 if 0 t 1, Λ(t) := 2 t if 1 t 2, and Λ(t) := 0 otherwise. Then we have the relations W C s (R n ) and dim H Γ(W ) = n + 1 s. Proof of Theorem The Theorem follows essentially by the same arguments given in the Proofs of Theorem and Corollary 4.4.7, but with V (j), H 0 (j), H(j) in place of V j, H0, j H j, respectively. In fact, by using an argument given in part (ii) of the Proof of Corollary 4.4.7, we may assume n = 1, and by using the argument given in part (i) of the Proof of Theorem 4.4.6, with elementary adaptations, we obtain W C s (R). In what follows we will prove the equality dim H Γ(W ) = 2 s (for n = 1). Consider j N, and let j be a square with width H(j) 1 and contained in a rectangle of the form [(k 1)H(j) 1, kh(j) 1 ] [(b j,k 1)V (j) 1, b j,k V (j) 1 ] see the remark that follows the Proof of Theorem Thus, by using that remark we obtain µ 0 ( j ) = H(j) 1 H(j) 1, so µ V (j) 1 0 ( j ) = H(j) 2 = H(j) s j (H(j) 1 ) 2 s j. Let B r be a ball centered at any P Γ(W ) with H(j) 1 B r = r H(j 1) 1, with j 2 integer. Hence, µ 0 (B r ) cµ 0 ( j 1 ) = c(h(j 1) 1 ) 2 s j = c(h(j) 1 ) 2 s j H 2 s j j. Consider δ > 0 and let j δ N be sufficiently large. Thus, for j j δ it holds µ 0 (B r ) c(h(j) 1 ) 2 s j H(j) (2 s j)δ c(h(j) 1 ) 2 s δ H(j) (2 s)δ = c(h(j) 1 ) 2 s (3 s)δ r 2 s (3 s)δ. By remark (b) of Definition we get dim H Γ(W ) 2 s (3 s)δ, δ > 0. On the other hand, we have also the estimation of the mass from below, µ 0 (B r ) cµ 0 ( j ) = c(h(j) 1 ) 2 s j = ch s j 2 j (H(j 1) 1 ) 2 s j, and it holds also lim j H j /H(j 1) δ = lim j (H j /H(j) δ 1+δ ) 1+δ = 0. Therefore, for 134

144 6.3 Comparison of different techniques 6 Disc. Applic. Open problems j j δ, we get also the inequalities µ 0 (B r ) c(h(j 1) 1 ) 2 s j H(j 1) (sj 2)δ c(h(j 1) 1 ) 2 s+δ H(j 1) (s 2)δ = c(h(j 1) 1 ) 2 s+(3 s)δ r 2 s+(3 s)δ. By remark (b) of Definition we get dim H Γ(W ) 2 s + (3 s)δ, δ > 0. Therefore, we have dim H Γ(W ) = 2 s and the Proof is complete. 6.3 Comparison of different techniques In Section 5.2 we proved the Theorem by using a wavelet approach for the Besov spaces see Lemma with Definition In order to prove that Theorem by using a frequency approach, we will need to use Lemma and the density Lemma 6.2.3, as well as the following Lemma (Cf. remark that follows the Proof of Lemma ) Lemma ([31], p. 22) Let Ω R n be a compact set. Let 0 < p q and α (N 0 ) n a multi-index. Then there exists c > 0 such that for all f L Ω p (R n ) we have D α f Lp (R n ) c f Lp (R n ). Lemma Let ϕ j as in Definition for all j N 0. Consider f S (R n ) and define f k := k j=0 ϕ j f for all k N. Then in S (R n ) with the strong topology we have Proof of Lemma lim f k = (2π) n 2 f. k Define b k := k j=0 F ϕ j, k N. Then by remark (b) of Definition 2.1.8, for all k N it holds b k (ξ) = 1 if ξ 2 k and b k (ξ) sup F ϕ for others ξ R n. Let ρ > 0 and A ρ := {ψ R n with p 2n,0 (ψ) < ρ}. In (S (R n ), S(R n )), we have < b k 1, ψ > ξ 2 k (sup F ϕ + 1) ψ(ξ) dξ for all ψ A ρ. Then the estimation sup ψ Aρ < b k 1, ψ > c ξ 2 k dξ ξ 2n holds. This last integral tends to 0 when k. Therefore in S (R n ) with the strong topology we have lim k b k = 1. On the other hand, by [22], pp. 34 and 38, with 135

145 6.3 Comparison of different techniques 6 Disc. Applic. Open problems elementary adaptations, we have in S (R n ) the identity F 1 1 = (2π) n 2 δ, where δ is the Dirac tempered distribution on R n. Then by the continuity of F 1 on S (R n ) we have, in S (R n ) with the strong topology, the convergence j 0 ϕ j = (2π) n 2 δ. By the continuity ([22], p. 18) of the convolution operator on S (R n ) we have with ( the strong topology the convergence lim k f k = j 0 j) ϕ f. Then the identity f = δ f in S (R n ) completes the Proof. The following Lemma concerns continuous real functions and follows by Lemmas and Lemma Consider α R. Let ϕ j be as in Definition for all j N 0. Let f : R n R be a continuous function with f S (R n ). Then we have (2π) n 2 f V α (R n ) Re(ϕ j f) V α (R n ). j 0 This inequality is a consequence of the following one, where I is a cube in R n as in Definition 2.2.2: (2π) n 2 osci (f) osc I Re(ϕ j f). j 0 Nevertheless, at least for 0 < α < 1 there is no lower (or upper) counterpart of the first inequality for V α (R n ) (or respectively V α (R n ) see Remark (b) for definition), because the right hand side always equals 0 in this case. (In fact, in this case we have ϕ j f S(R n ), see Lemmas 6.2.1/6.2.2.) Proof of Theorem (Frequency approach) By [31], p. 131, f is continuous, and as we can see at the beginning of the first Proof, given in Section 5.2, we can assume 1 p < in order to prove the Theorem. Let m N be such that supp f [ m, m] n, and define and j as general sums over dyadics I restricted to the cubes [ m, m] n and [ m2 j, m2 j ] n, respectively. (i) Consider j N 0 and ν N 0. Define G j := {ξ R n : ξ < 2 j+2 } and consider ψ j S G j (R n ). 136

146 6.3 Comparison of different techniques 6 Disc. Applic. Open problems Let I = 2 νn, I as in Definition with N = 2. (i1) Let ν j. We have osc I (Reψ j ) 2 sup I ψ j. Let j 0 N 0, then by Hölder s inequality and because ν j, we have sup ψ j (2 ν 2m) n p 1 p sup ψ j p I =2 νn I I =2 νn I 1 p p 1 νn c m 2 p Taking j 0 sufficiently large then by Lemma the factor ( ) 1 p above by c2 j 0 n p ψj (2 j ) Lp (R n ) = c 2 j n p ψj Lp (R n ). j sup I =2 j 0 n I ψ j (2 j ) p can be estimated from 1 p. Because ν j and γ n p then we have νγ + (j ν) n p jγ. Consequently, it holds 2 ν(γ n) I =2 νn osc I (Reψ j ) c m 2 jγ ψ j Lp(R n ). (i2) Let ν j. By the mean value Theorem we have osc I (Reψ j ) 2 ν n i=1 sup I x i Reψ j. Let i {1,..., n}, then by the Hölder s inequality we have sup p Reψ j I =2 νn I x i (2ν 2m) n p 1 p sup ψ j I =2 νn I x i Let ν 0 N 0. Because ν j then the last sum can be estimated from above by j sup I =2 (ν+ν 0 j)n I ( x i ψ j )(2 j ) p. Taking ν 0 sufficiently large, then by Lemma the factor ( ) 1 p can be estimated from ( ) above by c2 (ν+ν 0 j) n p ψ j (2 j ). x i Lp (R n ) By Lemma this last norm can be estimated from above by c 2 j ψ j (2 j ) Lp (R n ) = c2 j 2 j n p ψj Lp (R n ). 1 p. Because ν j then we have ν min{1, γ} ν + j j min{1, γ}. Then we have 2 ν(min{1,γ} n) I =2 νn osc I (Reψ j ) c m 2 j min{1,γ} ψ j Lp(R n ). (i3) Looking at (i1) and (i2) we conclude in particular, for all j, ν N 0, the inequality 2 ν(min{1,γ} n) I =2 νn osc I (Reψ j ) c m 2 jγ ψ j Lp (R n ). (ii) Let j N 0, ν N 0 and ϕ j as in Definition

147 6.3 Comparison of different techniques 6 Disc. Applic. Open problems We have ϕ j f L G j p (R n ). Therefore, by (i3) and the density Lemma we have 2 ν(min{1,γ} n) I =2 νn osc I Re(ϕ j f) c m 2 jγ ϕ j f Lp (R n ). (iii) Let ν N 0. By the second inequality of Lemma and by Definition we have 2 ν(min{1,γ} n) Of course, we may replace I =2 νn osc I (f) c m f B γ p,1 ). (Rn by in the obtained inequality. Then by taking the sup ν 0 on the left hand side we obtain the desired inequality. In Section 5.2 we proved the Theorems and by using a wavelet approach. Next, we state the Theorems 6.3.5/6.3.7, which are very similar to that Theorem 3.3.4, but now we use a frequency approach in the respective Proofs. Lemma (Young s inequality, see [30], p. 31, particular case) Let 1 p and consider f L p (R n ) and g L 1 (R n ). Then we have the inequality f g Lp(R n ) f Lp(R n ) g L1 (R n ). Next, we give a stronger version for the case 1 p < of Theorem In fact, the following Theorem takes advantage of the Young s inequality see Lemma 6.3.4, by dealing with a frequency approach that uses a version of compactly supported functions φ j instead of wavelets. Of course, this result is also deeper than Theorem for 1 p <, whose Proof deals with the usual ϕ j cutoff functions which are not compactly supported. Theorem Let 1 p < and γ R. Then we have the embeddings L p (R n ) V γp (R n ) B γ p, (R n ) and L 1 (R n ) V γp (R n ) B γ p, (R n ). Proof of Theorem (Frequency-wavelet approach) Let f V γp (R n ). Of course f is bounded and measurable, and then f S (R n ). Let ϕ and ϕ j as in Definition for all j N

148 6.3 Comparison of different techniques 6 Disc. Applic. Open problems (i) By Lemma we obtain the inequality ϕ 0 f Lp(R n ) c min{ f L1 (R n ), f Lp(R n )}. (ii) Consider ɛ > 0. Let φ S(R n ) be such that supp φ {x R n : x 1}, R n φ(t)dt = 0, and (F φ)(ξ) 0 for ξ R n with d(ξ,supp (F ϕ)) < ɛ. Here, d is the usual distance between points or sets in R n. It is easy to find this φ, because supp φ(k ) shrinks and supp F (φ(k )) enlarges when k enlarges. Hence, all we need is to replace φ by φ(k ) for k sufficiently large. Define φ j (x) := 2 jn φ(2 j x), x R n, j N, as well as τ : R n R with (F τ)(ξ) := (F ϕ)(ξ) (F φ)(ξ) if d(ξ,supp (F ϕ)) < ɛ and (F τ)(ξ) := 0 otherwise. Let j N. Because F ϕ = (F τ)(f φ) then we have also F ϕ j = (F τ)(2 j )(F φ j ). Therefore we have ϕ j = 2 jn τ(2 j ) φ j. By the associativity of the convolution on S (R n ) and by Lemma we obtain ϕ j f Lp (R n ) 2 jn τ(2 j ) L1 (R n ) φ j f Lp (R n ). Then ϕ j f Lp (R n ) c φ j f Lp (R n ). (iii) We notice here that, for j N, I as in Definition 2.2.2, x I, the hypothesis R n φ(t)dt = 0 justifies that it holds, see Definition (ii1/ii2) with ε = 0, the identity R n φ j (t)f(x t)dt = R n φ j (t)f I (x t)dt, therefore it holds also (φ j f)(x) = t 2 j φ j (t)f(x t)dt = t 2 j φ j (t)f I (x t)dt, so we have (φ j f)(x) t 2 j φ j (t)f I (x t) dt. The rest of the Proof consists in to applying the same calculation techniques as in part (ii) of the Proof of Theorem 6.3.7, putting j 0 = 1, γ = γ, ε = 0 and replacing ϕ j f by φ j f. In this way we obtain, for all j N, therefore sup 2 jγ φ j f Lp (R n ) c f V γp (R n ). j 1 2 jγ φ j f Lp (R n ) c f V γp (R n ), and (iv) By (ii) and (iii) we have sup j 1 2 jγ ϕ j f Lp(R n ) c f V γp (R n ). Therefore by using (i) and Definition we get f B γ p, (R n ) c max { min{ f L1 (R n ), f Lp (R n )}, f V γp (R n )}, in other words f B γ p, (R n ) c min{ f L1 (R n ) V γp (R n ), f Lp(R n ) V γp (R n )}. The Proof is 139

149 6.3 Comparison of different techniques 6 Disc. Applic. Open problems now complete. Definition Consider ϕ j as in Definition for all j N. Let f : R n R be a bounded and measurable function with a compact support. Consider j N and 0 < ε < 1. (i) We define (ϕ j f)(x) := t 2 j+[jε] ϕ j (t)f(x t)dt, x R n. (ii) (ii1) We define ( n i=1 [a i, a i + 2 j ]) # := n i=1 [a i 2 j+[jε], a i + 2 j + 2 j+[jε] ], where a i R, i = 1,..., n. (ii2) Let I = 2 jn, I as in Definition We define f I (x) := f(x) k I for all x R n, where k I := 1 f(t)dt. (Here, I # stands for the volume of I #, ruling I # I # out of the remark of Definition ) (ii3) Let I = 2 jn, I as in Definition 2.2.2, and consider the interior I o of I. We define almost everywhere (ϕ j f)(x) := t 2 j+[jε] ϕ j (t)f I (x t)dt, x I o. Remark: We have I # f I (t)dt = 0. Moreover, ϕ j f is continuous and ϕ j f is continuous almost everywhere. Lemma Consider 0 < p, 0 < q and s R. Let f : R n R be a bounded and measurable function with a compact support. Let j 0 N and 0 < ε < 1. Then see Definition we have the equivalences (modification if q = ) f B s p,q (R n ) < ( ) 2 js q ( ) ϕ j f Lp (R n ) < 2 js q ϕ j f Lp (R n ) <. j j 0 j j 0 Proof of Lemma Let R, L > 0 be such that Definition for all j N 0. supp f B R (0) and sup f L. Let ϕ and ϕ j as in By Lemmas 6.2.1/6.2.2 we have ϕ j f S(R n ), j N 0 and consequently it holds ϕ j f Lp(R n ) <, j N 0. Let k > 0. Then we have x k ϕ(x) c k, x R n. Consider j N sufficiently large. 140

150 6.3 Comparison of different techniques 6 Disc. Applic. Open problems (i) Let x R n. We have (ϕ j f)(x) = 0 for x R + 1 and (ϕ 2 j f)(x) = 0 for x R (ii) For all x R n with x R + 1 we have (ϕ j f)(x) c k,r,l 2 j(n k) x k. (iii) Let k n + 1 and x R n. Hence (ϕ j f)(x) (ϕ j f)(x) c k,l 2 j ε 2 (n k). (iv) Let k n + 1 and x R n. Then we have (ϕ j f)(x) (ϕ j f)(x) Lϕ t 2 j+[jε] j (t)dt = L ϕ t 2 j+[jε] j (t)dt L ϕ t 2 j+[jε] j (t) dt L t 2 j+j 2 ε ϕ j(t) dt L t 2 j 2 ε ϕ(t) dt c kl t 2 j 2 ε t k dt = c k,l 2 j ε 2 (n k) t 1 t k dt c k,l 2j ε 2 (n k). (v) Then by (i)-(iv), if we take k n + 1 sufficiently large we obtain the desired equivalences. The Lemma above allow us to prove the following Theorem by using a frequency approach, although this Theorem is actually a weak version of Theorems 3.3.4/ Theorem Let 0 < p < and γ > γ. Suppose that f V γ max{1,p} (R n ) has a compact support. Then we have f B γ p, (R n ). Proof of Theorem (Frequency approach) Of course f is bounded and measurable, and then f S (R n ). Let 0 < ε < 1. We will take into account Lemma with Definition (i) Let 0 < p 1 and j N. Let R > 0 be such that supp f B R (0). Then, analogously to part (i) of the Proof of Theorem 3.3.4, by Hölder s inequality we obtain x R+1 (ϕ j f)(x) p dx ( x R+1 ( (ϕ j f)(x) p ) 1 p dx ) p ( x R p dx ) 1 p. We have ϕ j f Lp (R n ) c R ϕ j f L1 (R n ), therefore we may assume 1 p <. (ii) Let 1 p < and consider f V γ p (R n ). Suppose j N, j j 0 with j 0 sufficiently large, and let I = 2 jn, I as in Definition Then for x I we have (ϕ j f)(x) ϕ j (t)f I (x t) dt c2 jn 2 ( j+[jε])n sup f I c2 jεn sup f I. t 2 j+[jε] I # I # Therefore we have sup I ϕ j f c2 jεn sup I # f I. By the equality of the remark that follows Definition 6.3.1, we have (sup I # f I ) (inf I # f I ) 0, and therefore 141

151 6.3 Comparison of different techniques 6 Disc. Applic. Open problems sup I # f I sup I # f I inf I # f I. The right hand side of the last inequality can be estimated from above by I osc I (f I ) = I osc I (f), where the last two sums are taken over all (1 + 2 [jε] 2) n cubes I I #, I = 2 jn, I as in Definition ( ) 1 Hence ϕ j f Lp (R n ) I =2 I sup jn I ϕ j f p p c2 jεn ( c2 jεn ( 2 jn I =2 jn sup I # f I p ) 1 p c2 jεn ( 2 jn I =2 jn ( I osc I (f)) p) 1 p 2 jn I =2 (1 + 2 [jε] 2) ) 1 np jn I (osc I (f)) p p. This last quantity can be estimated from above by ( c2 j2εn 2 ) 1 jn I =2 jn I (osc I (f)) p p c(2 sup f ) p 1 p 2 j2εn ( 2 jn I =2 jn I osc I (f) ( c(sup f ) p 1 p 2 j2εn 2 jn (1 + 2 [jε] 2) ) 1 n p I =2 osc jn I (f) c (sup f ) p 1 p 2 j3εn 2 j εn p ( 2 jn I =2 jn osc I (f) ) 1 p ) 1 p. On the other hand, because f L 1 (R n ) then we have inf f = 0, and therefore sup f f V γ p (R n ). In this way, 2jγ ϕ j f Lp(R n ) can be estimated from above by p 1 p c f V γ p (R n ) 2j3εn f 1 p = V γ p (R n ) c2j3εn f V γ p (R n ). Then by taking ε > 0 small we get sup j j0 2 jγ ϕ j f Lp (R n ) c f V γ p (R n ) <. So, by Lemma we have f B γ p, (R n ) < and therefore the Proof is complete. Comparing different approaches: So far, we have been acquiring experience and collected techniques in what concerns smoothness and oscillation or fractal dimensions of the graphs of continuous real functions. On the other hand, it is convenient to decide in each situation the more appropriate approach, comparing mainly the frequency and wavelet approaches to Besov- Triebel-Lizorkin spaces. By frequency approach we mean to consider the Definition for those spaces, whereas the characterization given in Lemmas 5.2.2/5.2.5 with 142

152 6.3 Comparison of different techniques 6 Disc. Applic. Open problems Definition based in the reference [37], p. 194, is an wavelet approach. We will compare, for each of the main results above dealing with that topic, these two approaches, which are somewhat different, at least at first glance. The Theorem which we proved in Section 5.2 by using the wavelet approach, or alternatively the Theorems 6.3.5/6.3.7, have as a practical intention to be used in order to prove the Corollary 3.3.5, and each one of these three results is sufficient for that. In fact, although the Theorem is not a sharp result, it is asymptotically sharp, and that is all we need to obtain an estimate for box dimension as in Corollary However, looking to these three Theorems and respective Proofs, we realize first that the Theorem is not only weaker than the other two Theorems, but also demands a much more complicated Proof. This is due to the fact that the functions ϕ j on the basis of the frequency approach recall Definition are not compactly supported. By using wavelets in the Proof of Theorem 3.3.4, which are compactly supported recall Lemmas 5.2.2/5.2.5 with Definition 5.2.1, the Proof comes much simpler, in particular we don t need to use the Lemma Nevertheless, as soon as we replace the functions ϕ j by another ones φ j with compact support, as in the Proof of Theorem 6.3.5, we recover all the advantages of the wavelets and additionally we can also apply classical results as the Young s inequality for 1 p see Lemma and comments following it in order to get a slightly stronger embedding. However, in order to apply those functions φ j we needed to introduce part (ii) in that Proof of Theorem If we are interested in trigonometric or exponential series with phases, then we can make a good characterization in terms of smoothness and box dimensions by using the frequency approach. This was made in Theorem 4.2.1, Remark 4.2.2, Theorem and Remark 4.2.4, taking advantage of the flexibility available in this approach concerning different phases in each frequency, which does not happen in the wavelet approach. We have exactly the same situation, in favor to the frequency approach, when we want to construct a particular series, representing a function whose graph has a box dimension above a prescribed value. This can be observed in the Theorems 3.4.1/3.4.4 with respective chirp functions as in Definitions 3.4.1/3.4.2, where we use 143

153 6.3 Comparison of different techniques 6 Disc. Applic. Open problems appropriately the flexibility in the choice of the positions 1/ ln j, m(j, l) or m(j, k), m(j, k, l). On the contrary, in the case of the other chirp function referred in Theorems 3.4.5/3.4.6 with Definition 3.4.3, we are interested on an estimation of the box dimension from above, so in this situation the wavelet approach can be an excellent alternative. In general estimations from above for upper box dimension, we have used frequently the Corollary In order to prove such an estimation we made use of the embedding given by Theorem 3.3.1, which we proved in Section 5.2 by using a wavelet approach for the Besov spaces. In order to prove that Theorem by using this approach we used Lemma instead of Definition and we did not need to use neither the Lemma nor the density Lemma By the way, actually we could not use that Lemma even if we have wanted to do it, because in the wavelet approach we do not have bounded frequence band-width. In fact, unlike in the frequency approach, where F (ϕ j f) are compactly supported functions recall Definition 2.1.8, in the wavelet approach the functions f j see the Proof of Theorem given in Section 5.2 are compactly supported themselves, and the price to pay for this is that F f j are not compactly supported. On the other hand, we deduced general estimations from below for box dimensions in Corollary and Remark (b). In the Proof of such an estimation we used Theorem 3.3.4, or alternatively the Theorems and 6.3.7, which we have already compared above. We notice also here that if F ψ 0 would be compactly supported, then in part (ii1) of the Proof of Theorem 3.3.4, given in Section 5.2, the term originated from j = 0, i.e., ( m Z < f, ψ n 0 ( m) > p) 1 p = ( m Z (ψ n 0 f)(m) > p) 1 p, could be estimated by c ψ 0 f Lp (R n ) by using the Lemma In that case we would obtain the same result as in Theorem by applying the Young s inequality as in Lemma 6.3.4, but in fact we cannot apply the Lemma in the wavelet approach, due to the frequency-unboundedness of the wavelet ψ 0. In what d-sets or h-sets are concerned, as we saw in Definition and remark following it, and in respective Theorem and Corollary 4.4.7, or even in Theorems 4.5.1/4.5.2, it is very convenient to make an approach by using non-smooth atoms, 144

154 6.4 Development of applications 6 Disc. Applic. Open problems according to the reference [36]. The same approach is also appropriate in order to construct graphs which are not h-sets, as have been done easily in Corollary and remark following it, according to Definition and Figure 14. In fact, as we saw in the explanation at the end of Section 4.4, as well as in the Proof of Theorem 4.2.7, if we want to use any kind of lacunary series of smooth type, then we should make a restriction of the function to a Cantor-like set. This is due to the overcharge of mass that occurs around the stationary points of the partial sum of the series, which is a difficulty that the Cantor-like set solves by removing short intervals around these points. Recall that we explored just the opposite situation in Corollary 4.4.4, where the local maximums of a triangular-type wave cause a lack of mass in the graph of the sum of the series. 6.4 Development of applications Fractals everywhere: There is a strong physical motivation to study fractals, since they appear in innumerable contexts, sometimes originated from turbulence phenomena with a statistic distribution. On the other hand, the fractal signature in the nature is frequently expressed in a deterministic way. Scaled structure of sand and gravel or of the buildings or roads and streets that compose a city, are examples of functional concretizations of fractal structures in innumerable dimensions. On animals, circulatory and respiratory systems carry blood and oxigen and lead it throughout the body by a sophisticated fractal ramification, and there are also analogous structures in the plants. Remarkable, the fractals are canonical tools to waste energy, because they have lack of resonance frequencies. Instead, they have a fine structure of reflections in their big boundary, to which is associated a resistive coefficient of dissipation. Namely, the trees and forests are perfect and idiomatic examples of this. They brake the wind in a soft and even process, and the rocks in breakwaters work in such a way against loud waves, since they tend to create a fractal stucture. Sophisticated technologies imitate these structures, as the glasses of a car, which have a fractal surface in order to spread and even the light. 145

155 6.4 Development of applications 6 Disc. Applic. Open problems In what concerns applications, we firstly recognise some academic ones related with Sections 4.4 and 4.5, concerning the importance of d-sets or h-sets as graphs of functions. In fact, the need for this kind of constructions can be seen e.g. in [36], pp , in [6], p. 61, or in [28], p.67, Remark 7.1. In what follows, we will find applications for more practical purposes. Some applications related with the embeddings of Section 3.3: We will now switch to practical applications and use a somewhat loose language, since in this part we are interested only on experimental facts, although they were suggested by precise theoretical results. Let us start by recalling the main embeddings of the Section 3.3, Theorems and 3.3.4/6.3.7, as well as Theorems 3.4.1/3.4.4 together with Theorem and Remark We may write versions of those results as applications, by interpreting the norm operators as sensors for signals. In order to do this, let us restrict attention to continuous real functions supported on the unitary ball B 1 (0). By adding a to the notations for the corresponding Besov and oscillation spaces, we mean that we are considering such a restriction. Application Let 0 < p < and s n. Then, B,s p p,1(r n ) V,min(1,s) (R n ). This result expresses that B,s p,1 is at least as sensible as V,min(1,s). Application Let 0 < p < and s R. Then, L max(1,p) (R n ) V,s max(1,p) (R n ) B,s p, (R n ) and moreover it also holds V,s max(1,p)+ε (R n ) B,s p, (R n ). This result says that V,s max(1,p)+ε is at least as sensible as B,s p,. Application Let 0 < p, q and 0 < s n. Then, there exists a chirp-type p function Λ : R n R such that Λ B,s ε p,q (R n ) and Λ V,ε (R n ), ε > 0. And, for chirp-type signals, V,ε is strictly more sensible than B,s ε. p,q 146

156 6.4 Development of applications 6 Disc. Applic. Open problems The problem of joint detection and classification, or even segmentation, of signals with added random noise (see e.g. Figure 19), is one of the most difficult and fundamental. Actually, in practice high level of correct detection can be at the expense of high computacional effort, so there is a need for a good balance between accurateness and robustness, having in mind the aims we proposed. Next, we will simulate numerically the Applications above applied to this topic, since they sugest promising tools to handle the problem of signal detection. Numerical simulation for samples with dimension 1024: Consider random samples of noise η with dimension k 0 = 2 e 0 = 1024 (e 0 := 10), generated under a standard gaussian probability. On the other hand, let ζ be a sampling of a signal, with the same dimension k 0 = 1024, for instance a wave or a chirp as W ave signal := (A S cos(2π 0.01 k)) k 0 k=1, Chirp signal := ( A S cos(2π k 2 ) ) k 0 k=1, where A S is the amplitude of the signal, in our case a number between 0 and 1. Figure 19: Signal with added standard gaussian noise; sample dimension k 0 = 1024 In both cases, wave and chirp, we will run A S over all set { 1 500,..., }, getting then a cloud of 500 points, as shown in Figures 20 and 21 below. Each point of the cloud represents a signal, the abcissa is the amplitude A S of the signal and the respective ordinate is the statistical significance S S observed when the statistical test is one of the 147

157 6.4 Development of applications 6 Disc. Applic. Open problems operators Ψ F ou or Ψ Osc, according to the definitions below. The higher the point in the cloud, the higher the probability of being detected. We present also, from a rough interpretation of the cloud, the smallest signal we expected to detect on each case. The values of S S were estimated by the Monte Carlo method, in which we obtained a random sampling with dimension It is remarkable that in the Fourier case the computation time observed was about 55 times the observed for the oscillation case, so we expect much less computational effort for signal detection in the later case. We define the Fourier approach operator, based on (1.5) of [27], as 100 k 0 Ψ F ou := (η k + ζ k )e i2π m 200 k. m=0 In what concerns the oscillation approach consider, for each ν {1,.., e 0 }, u=1 k=1 2 ν 1 Osc(ν) := sup(η k + ζ k ) inf(η k + ζ k ), k ν,u k ν,u where for any pair (ν, u) both supremum and infimum are taken over all k ν,u {1 + 2 e 0 ν+1 (u 1),..., 2 e 0 ν+1 u}, i.e. k ν,u runs over all integers on that interval. In this way, we define the oscillation approach operator Ψ Osc := (1,.., e 0 ) (ln Osc(e)) e 0 e=1 (1,.., e 0 ) 2 (ln Osc(e)) e 0 e=1 2, by means of a scalar product in the numerator. What we can conclude by observing the Figures 20ab below is that a wave-signal is more likely to be detected by using the Fourier approach than the oscillation approach. On the other hand, Figures 21ab express that at least for these concrete chirp-signals (we have not tried others) the oscillation approach is more effective. 148

158 6.4 Development of applications 6 Disc. Applic. Open problems Figure 20a: Wave-signal detection - Fourier approach (Runovski, [27]) S S A S Figure 20b: Wave-signal detection - Oscillation approach S S A S 149

159 6.4 Development of applications 6 Disc. Applic. Open problems Figure 21a: Chirp-signal detection - Fourier approach (Runovski, [27]) S S A S Figure 21b: Chirp-signal detection - Oscillation approach S S A S 150