Journal of Function Spaces
Volume 2019 (2019), Article ID 4358261, 14 pages
https://doi.org/10.1155/2019/4358261

Research Article

On the Baire Generic Validity of the -Multifractal Formalism in Besov and Sobolev Spaces

Moez Ben Abid,¹ Mourad Ben Slimane,² Ines Ben Omrane,³ and Borhen Halouani²

¹Université de Sousse, Ecole Supérieure des Sciences et de la Technologie de Hammam Sousse, Sousse, Tunisia
²King Saud University, Department of Mathematics, College of Science, P.O. Box 2455, Riyadh 11451, Saudi Arabia
³Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia

Correspondence should be addressed to Mourad Ben Slimane; mbenslimane@ksu.edu.sa and Ines Ben Omrane; imbenomrane@imamu.edu.sa

Received 13 February 2019; Revised 7 May 2019; Accepted 16 June 2019; Published 29 August 2019

Academic Editor: Stanislav Hencl

Copyright © 2019 Moez Ben Abid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the spectrum of a function from the knowledge of the oscillation exponent of . The spectrum is the Hausdorff dimension of the set of points where has a given value of pointwise regularity. The oscillation exponent is measured by determining to which oscillation spaces (defined in terms of wavelet coefficients) belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the multifractal formalism.

1. Introduction

Multifractal analysis is concerned with the pointwise regularity and the scaling behavior of functions. It gives a powerful classification tool in various domains. In the setting of the classical Hölder pointwise regularity, it has been successfully used theoretically in (see [1–18] and references therein) and practically for signal and image processing (see [18–24] and references therein). Several related generic results, in the sense of Baire’s categories [15, 25–34] and in the prevalence sense [10, 11, 35, 36], were proved. Recall that a prevalent result is large in a measure sense, whereas a Baire generic result holds in a residual set (any countable intersection of open dense sets) in a well chosen topological vector space. Both Baire and prevalent generic sets are dense and stable by translation, dilation, and countable intersection. However, prevalence and Baire genericity usually differ widely. In , prevalence coincides with Lebesgue almost everywhere, and there exist subsets of with vanishing Lebesgue measure, but Baire generic. In infinite-dimensional spaces, there are stronger results of this type in [37, 38]. Nevertheless, in [39], Kolàr proved that the so-called HP-notion of genericity yields both Baire and prevalent results (see also [40]).

The Hölder regularity has some limitations (see [41, 42] and references therein). Hölder regularity is only defined for locally bounded functions. It can not take negative values. It is not stable under some pseudodifferential and integral operators. It is not significant in fractal boundaries where it takes only two values and . For instance, in fully developed turbulence, velocity is not bounded near vorticity filaments and yields negative singularities [20]. The same holds for microcalcifications in mammography [20]. In order to overcome these weaknesses, Hölder regularity was replaced by the pointwise regularity introduced by Calderón and Zygmund in [43] for functions that belong locally to to better study elliptic partial differential equations. This notion has recently been put forward in the mathematical literature in [42, 44–46].

Definition 1. Let . Let be a real number and . A function in belongs to if there exists and a polynomial of degree less than (with if ), such that

for some constant independent of .
The pointwise regularity of at is

Definition 1 written for corresponds to the Hölder regularity.

Definition 1 can be also extended to the case , where we consider the Hardy space instead of [47]. In [48] Theorem 1 p. 4, it is proved that , is decreasing, and is concave on . In [43], it is shown that, contrary to the Hölder regularity, the pointwise regularity for is invariant under pseudodifferential operators of order 0.

The -sets of are given by

The spectrum of is defined by the function

where denotes the Hausdorff dimension. By convention .

In [11, 49], Fraysse has computed the spectrum for almost every function, in the prevalence setting, in a given Sobolev or Besov space.

In [45, 46], Jaffard and Mélot have shown that the pointwise regularity is well adapted for fractal interfaces. They have also proved that if belongs to the Besov space for an , then the pointwise regularity is characterized by some conditions bearing on the moduli of the wavelet coefficients [47] (the definition of Besov is recalled in the next section). Note that (see [50])

Let us recall the result obtained in [46]; let be either the Daubechies [51] compactly supported wavelets in (where is the uniform Hölder regularity of ) or the Lemarié-Meyer [50, 52] wavelets in the Schwartz class of rapidly decreasing functions (we will write ), such that the family , for and , form an orthogonal basis of (note that we choose the normalization, not ). We will omit the letter and the summation with respect to . This will not affect the results of this work. The wavelets will be indexed in terms of dyadic cubes

so that we can write

For , set

Put

Let be the space of tempered distributions (i.e., the dual of ; let . Using the notation , the wavelet coefficient of is given by

If then

And

Recall that, for , the exponent of is given by

Remark 2. In [48] Section 3.4, it is proved that if , then does not depend on the chosen wavelet basis as long as (where ). When , if , then and if then .

For , the wavelet leader of at was introduced in [46]

where the sum is over all such that with . The sum (14) is finite if .

For , denote by the unique cube at the scale that contains and the set formed by the cube and all its adjacent cubes at scale . Let . Suppose that and (resp., and ). If , then

and see [48], Section 3.3 (resp., [46] Corollary 1 p. 553).

Remark 3. Formula (15) was used in [48] to extend the definition of the pointwise regularity to the case under the sole condition (see [48], Section 3.4). From now on, the pointwise regularity is defined by formula (15) for .

Using characterization (15), a multifractal formalism associated with pointwise regularities was conjectured by Jaffard and Mélot in [46]. Let us recall it.

Definition 4. For , , the oscillation space (see [46]) is the space of tempered distributions that satisfy

The left-hand term defines the -seminorm.

It is independent of the choice of the smooth enough wavelet basis or in the Schwartz class (see [53], Section 3.2).

This space together with global notions (3) and (4) can also be defined locally; let

For , set

and

Clearly

For and , set

Definition 5. For , , the oscillation space is the space of tempered distributions that satisfy

Remark 6. For this space, we can assume that functions and wavelets are compactly supported, with support that contains . Then, there exists , such that, at each scale , there are at most dyadic cubes for which does not vanish (see [17] in the proof of Proposition 10 p.36).

For , the local oscillation exponent of on is given by

The oscillation exponent of is defined as

The multifractal formalism in [46] states that

The local multifractal formalism on states that

These formalisms yield upper bounds valid for any function (see either [46], Theorem 2, p. 561, or [53], Section 3.2); for , define the upper -set of by

and the upper spectrum by

Define the local upper -set of by

and the local upper spectrum on by

Then

and

where

As in (26), we state the multifractal formalism for upper -sets as

We also state the local multifractal formalism for upper -sets on as

In [53], Leonarduzzi et al. have validated the multifractal formalism for some synthetic images and signals that include independent realizations of random processes.

This paper is devoted to the study of the Baire generic validity of both local and global multifractal formalisms in a given Sobolev or Besov space. Recall that Besov and Sobolev spaces are complete metrizable spaces [54]. Note that, if then for all . On the contrary, if and (resp., ), Jaffard and Meyer [55] (resp., [28]) have proved that a function in can be infinite on a dense set and thus nowhere Hölder regular. Similarly, if , there exist functions in the usual Sobolev space

which are everywhere locally unbounded [55].

In this paper, we are interested in the Baire generic validity of the multifractal formalisms in Besov spaces for and Sobolev spaces for and , under the condition . In the next section, we recall some tools from the theory of Besov spaces. We also add some embeddings between oscillation spaces and a relationship with the space BMO of functions of bounded mean oscillation. In the third section, we prove embeddings between Besov and oscillation spaces. We deduce a general lower bound for the oscillation exponent. In the fourth section, we show that the obtained lower bound is actually equality generically, in the sense of Baire categories, in a given Sobolev or Besov space. In the fifth section, we investigate the Baire generic validity of the multifractal formalisms. Finally, in the sixth section, we deduce a conclusion on the range of Baire validity of both the multifractal formalism and the multifractal formalism for upper sets.

All generic results are studied locally on () and also globally on .

Remark 7. Since only wavelet leaders for are needed in the values of pointwise regularity and the exponent, then, from now on, we will identify functions that have the same wavelet coefficients for .

2. Besov, Sobolev, and Oscillation Spaces

2.1. Besov and Sobolev Spaces

Let us first recall some properties of Besov spaces (for details, see, for example, [54, 56, 57]). Given a function , its Fourier transform is denoted by . Let with

For , let

Then the support of is included in the annulus and

is a partition of the unity. The Littlewood Paley definition of Besov spaces is the following.

Definition 8. Let , . Then

where

Note that is a norm (quasinorm when or ) on .

Besov spaces do not depend on the choice of . They are separable when both and are finite.

Let us recall their wavelet characterizations; for a given sequence of scalar numbers (where is as in (9)), let

with the usual modification when or (i.e., and the sums of th or th powers replaced by suprema over the same sets of indices), and

Then, for the above wavelets if either or is large enough ( in [57], Theorem 1.64, p. 48, and ( if , and if ) in [50, 58]), then

is an isomorphic map from onto .

Under the same condition on , it is also proved that, for and , the Sobolev space given in (36) is characterized by

where denotes the characteristic function of the cube .

Note that characterizations (44) and (45) do not depend on choice of the wavelet basis (see [50]).

Besov and usual Sobolev spaces are closely related (see [12, 59]):

and

The following embeddings hold (for example, see [60], Proposition 2.6, p. 245, Proposition 2.8, p. 245, and Theorem 2.14, p. 248, respectively):

The following interpolation property holds:

Using the isomorphic (44), we obtain similar embeddings between Besov spaces.

2.2. Oscillation Spaces

Proposition 9. The following embeddings between oscillation spaces hold:

and

We also have

And

and

If is as in (17), then the following local embedding holds:

Proof. Embeddings (52) (resp., (53)) follows directly from (49) (resp., (50)). Embedding (54) follows from (51) and the equivalence

Let us now prove embedding (55). Write the wavelet leader of at a cube (given in (14)) as

where

If , then

We know that, by Cauchy-Schwartz inequality, if and , then

If then ; then relation (63) applied to (62) with and (with ) yields

Then

It follows from (60) that

Put

Then, as in (64), relation (66) yields

This achieves the proof of embedding (55).
Embedding (56) follows from the fact that .
With regard to (57), for , , and ,

It follows that

Thanks to our (resp., Meyer ) normalization (7) for wavelets and (10), the right-hand term in (70) coincides with the Carleson condition for the wavelet characterization of the space given in Theorem 4, p. 150-151 in [50].
Finally result (58) follows from Remark 6 and the Hölder inequality:

Oscillation spaces were defined in terms of wavelet coefficients. They can be characterized by some Littlewood Paley conditions; moreover wavelet leaders can also be replaced by local norms [61].

When , the wavelet leaders of given in (14) boil down to the classical wavelet leaders

used for the characterization of the Hölder exponent. The corresponding oscillation spaces have been studied in [17, 62, 63] and denoted by . Their characterizations by differences were obtained in [62]. Their independence of the chosen wavelet basis in the Schwartz class was obtained in [17]. For either or , it is proved in [63] that these spaces are a variation on the definition of Besov spaces. On the contrary, the spaces for cannot be sharply imbedded between Besov spaces and thus are new spaces of really different nature.

Generalized oscillation spaces were also introduced in [46]; if (resp., ), then belongs to if its fractional derivative (resp., primitive) or order which we denote by (i.e., such that ) belongs to . Spaces allow the computation of fractal dimension of graphs and yield a multifractal formalism for chirp-type Hölder singularities that behave like (see [62]).

3. General Lower Bound on the Oscillation Exponent

The following general lower bounds of both local and global oscillation exponents hold.

Theorem 10. Let be any function in either the Besov space for or the Sobolev space for and . Then, for all such that ,

and

Proof. For , define the local Besov space (resp., Sobolev space ) as in (44) (resp., (45)) with replaced by (resp., replaced by ).

Result (73) will follow from (23) and the following proposition. Result (74) will be deduced from (24).

Proposition 11. Let . The following embeddings hold for all such that :

and

Embeddings between similar local Sobolev and oscillation spaces also hold (using (46) and (47)).

Proof. Let for and such that .
Assume that . If is as in (61), then using the property

we get

(a)If , then by Remark 6, the quantity given in (67) satisfies

Clearly

Since , then . There exists such that

It follows that for all

Since , then

Thus

(b)Let now .

It follows that

From 1.(a) since , then . Then

By taking , we deduce that

Thus

Assume that . Since , then there exists such that . It follows from the fact that that . Let and . Since write .

 Since , then . Since , then from above 1.(a) yields

On the other hand, since then . In fact

Since , then from (81) we get

Since then

It follows from (60) that

Then

Therefore

Using (54), both (90) and (96) yield

When tends to 0, fraction tends to 0 too, and tends to 1. Hence (76) holds.

4. Generic Oscillation Exponent

For a given Besov or Sobolev space, the residual set that we will construct will be generated from a saturating function (i.e., for which the lower bounds obtained in Theorem 10 become equality). Thanks to embedding (46), we can choose for saturating function for the Sobolev space the one obtained for the Besov space .

4.1. Saturating Function

For , let be the unique integer given by the irreducible representation

For , put

The following wavelet series will be called a saturating function:

where

and .

Remark 12. If and then . Both and share the same . The previous function satisfies

This yields

It follows that

It is easy to show that (see [15], Proposition 2, p. 532). Relation (102) yields that . Note that the norm of the local Besov space is as the global one but with restriction on

Therefore and .

Theorem 13. Let and be such that . Then the saturating function satisfies

Proof. Thanks to both (104) and Theorem 10, it suffices to show that

We know that, for ,

It is easy to show that

(the proof is the same as in [15], Proposition 4, p. 540-541, and we do not need the assumption ).

The following result estimates the wavelet leaders of and allows the computation of its local oscillation exponent on .

Proposition 14. Let and such that . Let be as in (101). Then there exist two positive constants and such that

Proof. For , either or . There exists such that, at each scale , there are subdyadic cubes of that have the irreducible representation . Then

The left-hand series corresponds to if .
Consequently, there exists such that

with

and

Since , we get

where means that the left quantity is bounded from below and above by positive constants times the right quantity.
It follows that

Hence (110) holds.

Relation (110) together with (23) yields (107).

4.2. The Residual Set

We will need the following lemma.

Lemma 15. Let and such that . Then, for all , for all ,

where .

Proof. Let and . Write instead of . Clearly

Let and .

(i)Suppose that . Then

 Using (118)

 Then

 where . We therefore get

(ii)If and , then

 Since , we get

 where . Then

Therefore (117) holds with .
Hence Lemma 15 holds.

Theorem 16. Let (resp., and ) and such that .

(1)For all , there exists a residual set of (resp., ) such that for all

and

(2)There exists a residual set of (resp., ) such that for all

 and

 Relations (126) and (127) also hold on .

Proof.

(1)(i) First consider . By Theorem 10, it suffices to prove the upper bound in (126). Thanks to embedding (46), it suffices to write the proof in Besov spaces. We will follow the idea of [64] in the construction of the residual set since it does not depend on the separability of the space . Let be the saturating function defined in (101). From now on, when it is necessary, we will make the dependency on the function in the previous notations (for example, we write instead of in (10), instead of , and instead of ). If , set

 where is as in (105). Clearly

 Write instead of . In [64] (Proof of Lemma 2.9, p. 1520-1521), it is proved that there exists such that

 and so, if , then the set is dense in . Let be a sequence of positive numbers which converges to . Put

 Then

 is a residual set of . If , then the following trivial relation holds:

 Choose

 where is as in Lemma 15 and is any constant satisfying (110). Let be the residual set given by (141) associated with the sequence . Let . Then for infinitely many integers , there exists such that . By (135)

 By Lemma 15 and Proposition 14,

 It follows from (131) that

 Since (139) holds for infinitely many scales and for all , and thanks to the fact that the in is actually a limit, we deduce that

 Result (106) allows achieving the proof of Theorem 16.(ii)Now if , it suffices to replace by defined in (101) and by . This means that the residual set of on which (126) will hold is

 where

 and

(2)Clearly

 is a residual set of on which (128) and (126) hold (using (24) and the first result in this theorem).

5. Generic Validity of the Multifractal Formalisms

We first show that condition in both result (15) and upper bound (31) holds for all functions in Besov spaces for and Sobolev spaces for and , for all such that .

Lemma 17. Let be any function in either the Besov space for or the Sobolev space for and . Then, for all such that ,

Proof. Embedding (56) implies that . It follows that . Since , then Theorem 10 yields .

Lemma 17 together with both Theorem 10 and upper bound (31) yields the following corollary.

Corollary 18. Let be any function in either the Besov space for or the Sobolev space for and . Then, for all such that

The following result shows that the Baire generic -spectrum is the same as the prevalent generic one obtained in [49].

Theorem 19. Let (resp., and ) and such that .

(1)For all , for all functions in the residual set of (resp., ) constructed in Theorem 16,

 and

(2)For all functions in the residual set of (resp., ) constructed in Theorem 16,

 and

 Both (147) and (148) also hold on .

Proof. As in the proof of Theorem 16, it suffices to give the proof for ; let . Relation (139) is satisfied for infinitely many scales . Let . For each such scale , write if satisfies (98) with . Let be the set of points that belong to , for an infinite number of values of the above ’s. In [15], p. 535-536, we can find the following result (see also [65]).

Proposition 20. For , the Hausdorff dimension of is . Moreover, there exists a -finite measure carried by such that for any with .

The following proposition bounds the pointwise regularity of on .

Proposition 21. Let (resp., and ) and such that . For all , for all ,

Proof. Let . Result (15), relation (139), and result (145) imply that

Put . Take if (resp., if ). Then if (resp., if ). Using Proposition 21, for all , . It follows that . Therefore, for all ,

Thus, by (20) and (146), for all ,

For , consider the set , where . Clearly . By (150), for all , . Then, by Proposition 20, . Using the additivity of the measure , we get .

Since , it follows that . Thus,

By (150), for , we get

For , . Then for all , . Thus, for , .

Both (147) and (148) also hold on . This achieves the proof of Theorem 19.

6. Conclusion

The following theorem summarizes the range of Baire validity of both multifractal formalism and multifractal formalism for upper sets, locally on the and also globally on . It follows directly from both Theorems 16 and 19.

Theorem 22. Let (resp., and ) and such that . Baire generically in (resp., ), for all , the multifractal formalism is valid for , and the multifractal formalism for upper sets is valid for all .
The same results hold for the local multifractal formalism on .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Mourad Ben Slimane and Borhen Halouani extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RG-1435-063. Mourad Ben Slimane is grateful to Stéphane Jaffard for his interest and for stimulating discussions.

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