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

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 thatfor 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 functionwhere 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 Thenand 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 withFor , letThen the support of is included in the annulus andis a partition of the unity. The Littlewood Paley definition of Besov spaces is the following.

Definition 8. Let , . ThenwhereNote 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 ThenIt follows from (60) that Put Then, as in (64), relation (66) yieldsThis 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 propertywe get(a)If , then by Remark 6, the quantity given in (67) satisfiesClearlySince , then . There exists such that It follows that for all Since , thenThus(b)Let now .It follows thatFrom 1.(a) since , then . ThenBy taking , we deduce thatThus 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) yieldsOn the other hand, since then . In fact Since , then from (81) we getSince thenIt follows from (60) thatThenTherefore 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 representationFor , put

The following wavelet series will be called a saturating function: whereand .

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