#### Abstract

The classical Hölder regularity is restricted to locally bounded functions and takes only positive values. The local regularity covers unbounded functions and negative values. Nevertheless, it has the same apparent regularity in all directions. In the present work, we study a recent notion of directional local regularity introduced by Jaffard. We provide its characterization by a supremum of a wide range oriented anisotropic Triebel wavelet coefficients and leaders. In addition, we deduce estimates on the Hausdorff dimension of the set of points where the directional local regularity does not exceed a given value. The obtained results are illustrated by some examples of self-affine cascade functions.

#### 1. Introduction

Multifractal analysis describes geometrically and statistically the distribution of pointwise regularities (or singularities) of irregular functions on . It was first introduced in the context of the statistical study of fully developed turbulence in the mid 80’s [1]. The classical notion of pointwise regularity most commonly used is Hölder regularity. Let and . Let denote the open ball of radius , centered at . Recall that a function on belongs to if there exist constants and and a polynomial of degree at most the integer part of , such thatClearly, this condition makes sense only if is locally bounded. It is equivalent toIt describes how the norm of , properly renormalized by substracting a polynomial, behaves in small neighborhoods of .

The Hölder exponent of at is defined asThe multifractal formalism relates some function spaces norms of to the Hölder spectrum (which is the Hausdorff dimension of the set of points where has a given Hölder exponent). The idea of using wavelets in multifractal analysis has been worked out first by Arneodo et al. [2]. Wavelets characterize Hölder regularity and many function spaces.

In many PDE’s, the natural function space setting is for or a Sobolev space which includes unbounded functions. In such cases, one replaces the norm in (2) by the norm. This corresponds to the following weaker condition introduced by Calderón and Zygmund [3].

*Definition 1 (let ). *Let be a function which belongs locally to .

If , one says that if there exist constants and and a polynomial of degree at most such thatIf , one says that if there exist constants and such thatThe exponent of at is defined as

Clearly if then . The usual Hölder regularity , for , corresponds to . If , then for all . If then for all .

In [4, 5], Jaffard and Mélot proved that if is a domain of and is on the boundary of , then by taking or , the condition for the characteristic function of coincides with the bilaterally weak accessibility of at . Let . Recall that is called bilaterally weak accessible at if there exist two constants and such thatwhere is the Lebesgue measure. This remark allows performing a multifractal analysis of fractal boundaries [5, 6]. This analysis has many applications in physics, mechanics, or chemistry where many phenomena involve fractal interfaces. More pointwise exponents for classification of the geometry of fractal boundaries were studied by Jaffard and Heurteaux [7]. The relationship of these exponents to local dimension computation was proved by Tricot [8].

In [4, 5], a wavelet characterization of the regularity was obtained. Moreover, an associated multifractal formalism was conjectured: the Hausdorff dimension of the set of points , where the exponent of a function is equal to , may be derived from some global functional norms extracted from .

Unfortunately, if then Definition 1 has the same apparent value in all coordinate directions. However many signals belong to classes of functions with various directional regularity behaviors (see, for instance, [9–17] and the references therein). These behaviors are important for detection of edges, efficient image compression, etc. (see, for instance, [12] and the references therein). Classical (isotropic) wavelets are not optimal for analyzing directional or anisotropic features like edges. Extensions of wavelet bases which can be elongated in particular directions were considered. Candes and Donoho [18] and Mallat [19] (resp., Donoho [20] and Guo and Labate [21]) have used ridgelets and bandelets (resp., wedgelets and shearlets) to detect singularities along lines and hyperplanes (resp., discontinuities along smooth edges). Sampo and Sumetkijakan [16, 22, 23] (resp., Jaffard [24]) have used curvelets and Hart-Smith transform (resp., the anisotropic Gabor-wavelet transform) to detect directional pointwise Hölder singularities.

Many signals present anisotropies quantified through regularity characteristics and features that strongly differ when measured in different directions [13, 14, 24–29]. Several authors were concerned with the problem of obtaining useful decompositions of isotropic and anisotropic function spaces in simple building blocks: atoms, quarks, wavelets, splines (see, for example, Farkas [30], Garrigós et al. [31, 32], Hochmuth [33], and Kamont [34]). Triebel family of anisotropic wavelets yield characterizations of anisotropic Besov spaces [35, 36] and anisotropic Hölder regularity (see Ben Slimane et al. [37–39]).

Let be a unit vector. To take into account pointwise directional behavior in direction , it is natural to define the local regularity at a point in direction as the exponent at 0 of the one variable function , that is,We wish to derive or estimate the Hausdorff dimension of the set of points where from some global quantities extracted from the function itself and not its traces (actually these traces are unknown since corresponding points are unknown). One cannot expect directional local regularity to be characterized in terms of the size of the wavelet coefficients by taking wavelets on because is defined as the trace of on a line, which is a set of vanishing measure and wavelets on have a support of nonempty interior. Thus one should take into account the values of around the line considered. Therefore the definition of directional local regularity should include such information. However, in the asymptotic of small scales, the values taken into account should be localized more and more sharply around this line. These considerations motivated the following definition of Jaffard [24].

*Definition 2. *Let be an -uple of nonnegative real numbers. The average regularity is the harmonic mean of the , i.e.,The anisotropy indices areLet be an orthonormal basis of . Denote by the coordinates of on the basis .

SetThen is a quasi-norm on ; i.e., it satisfies the requirements of a norm except for the triangular inequality which is replaced by the weaker conditionIt is also continuous and homogeneous in the sense thatThe corresponding -ball of -radius centered on is a rectangle with sides parallel to the axes of coordinates, centered at and with side-length in the -direction.

Let be a function which belongs locally to . One says that if there exist constants and a polynomial of degreein the sense thatsuch that

*Remark 3. *Actually, here replaces the elongated ellipsoid of axis of lengths , centered on , considered in [24]. This ellipsoid corresponds to the ball centered at with radius , where is the quasi-norm on defined by , and for , is the unique value of for which . Both and are equivalent. Nevertheless, has the advantage of being easier in the computations of distances.

In the isotropic case ( for all ), coincides with the Euclidean norm.

We first extend the definition of for an -uple of nonpositive real numbers as follows.

*Definition 4. *Let be an -uple of nonpositive real numbers. We say that if there exist constants and such that

Contrary to the case studied in [24], there is no partial ordering property for ; i.e., if either or then does not imply that . Indeed, the harmonic mean values of and of satisfy , but the corresponding anisotropy indices and are not necessarily the same. Nevertheless, we will show that the following substitute for directional local regularity given in (8) makes sense (see next section).

*Definition 5. *Let be a unit vector and be any orthonormal basis starting with the vector . Let be a function. SetandIf , define the exponent of at in direction byIf , define the exponent of at in direction by

*Remark 6. *Clearly we can choose any orthonormal basis starting with the vector , in fact the component is the same in any and is equal to the inner product of with , and all norms of are equivalent.

*Definition 7. *Let be a unit vector. Let be a function. The directional spectrum (resp., upper spectrum) of in direction is the map which associates with each the Hausdorff dimension (resp., ) of the set of points where (resp., ).

The paper is organized as follows. In Section 2, we first show that Definition 5 makes sense. Then we prove a criterion of directional pointwise exponent in terms of a supremum on a wide range of oriented anisotropic regularities. We therefore deduce a first upper bound for the directional spectra. In Section 3, we establish criteria of directional pointwise exponent in terms of decay condition on Triebel anisotropic wavelet coefficients and wavelet leaders. In Section 4, we obtain an alternative upper bound for the directional spectra expressed as an infimum of Legendre transforms of some anisotropic scaling functions. Finally, in Section 5, the obtained results are illustrated by some examples of self-affine cascade functions.

#### 2. Directional Regularity Criterion

##### 2.1. Approval of Definition 5

We first show that Definition 5 makes sense.(i)Suppose that with . Let . Set . Then and . Set and . We will prove that ; clearly, the harmonic mean values of and of satisfy . The corresponding anisotropy indices are the same.Since then there exist constants and a polynomial of degree less than (see (14) and (15)), i.e.,such thatSplit as the sum of two polynomials and , where the indices of the nonvanishing coefficients of satisfy . Therefore the indices of the nonvanishing coefficients of satisfy both and . Clearly and if then . For each index of the nonvanishing coefficients of It follows that .(ii)Suppose that with . Let . Set . Then and . Clearly the harmonic mean values of and satisfy . The corresponding anisotropy indices are the same. Since then there exist constants and such thatIt follows that .

##### 2.2. Criterion of Directional Pointwise Exponent

We will establish a criterion of directional pointwise exponent in terms of a supremum on a wide range of oriented anisotropic regularities. The latest regularity is reminiscent of [40, 41] where Calderón and Torchinsky on one side and Folland and Stein on the other side have developed a theory of anisotropic spaces using the quasi-norm of Remark 3.

Let be such thatWe will say that is an anisotropy vector.

If , set . If , is a polynomial, define its -homogeneous degree by

*Definition 8. *Let be a function which belongs locally to . Let be an orthonormal basis of .

If , we say that belongs to if there exist constants and a polynomial of -homogeneous degree less than such thatIf , we say that belongs to if there exist constants and such that

Definitions 2 and 4 are related to the previous anisotropic regularity in setting. Let be an -uple of either nonpositive or nonnegative real numbers. Let (resp., ) be the corresponding harmonic mean value as in (9) (resp., anisotropy indices as in (10)).

If is an -uple of nonpositive real numbers then it follows from (29) and Definition 4 thatIf is an -uple of nonnegative real numbers thenIt follows from (14) and (15) thatWe deduce that

The following theorem extends the result obtained for the directional Hölder regularity in [38] to the setting. It has the advantage to cover unbounded functions and negative values.

Theorem 9 (let ). *Let be a unit vector and be any orthonormal basis starting with the vector . Let be the set of all anisotropy vectors with .**Let . Then the exponent of at in direction is given bywhere is the so-called anisotropic regularity of at oriented in basis , defined as*

*Proof (let ). *Assume that . Take and for all take . Set . Clearly, the anisotropy indices of coincide with .(i)If thenSince with , then it follows from Definition 5 thatThenTaking at the right the supremum over yields the lower bound in (34).(ii)If thenSoWe deduce that (36) holds too.Since with , then it follows from Definition 5 thatThenTaking at the right the supremum over yields the lower bound in (34).Let us now prove the upper bound.(i)Assume that with , and . It follows from (30) thatThenThusWhence Theorem 9 holds.(ii)The case where with with and is similar.

##### 2.3. General Upper Bound for the Directional Spectra

Let be the set of anisotropy vectors given in Theorem 9. Let be an orthonormal basis of . Define the --sets of in basis byand the upper -sets of in basis byTheorem 9 yields the following general upper bound for the directional upper spectrum given in Definition 7.

Corollary 10. *Let be as in Theorem 9. Let be a unit vector and be any orthonormal basis starting with the vector . If then the directional spectra of in direction given in Definition 7 satisfywhere denotes the Hausdorff dimension.*

#### 3. Criteria of Directional Regularity in Anisotropic Triebel Wavelet Bases

For each there are Daubechies [42] real valued compactly supported father and mother wavelets and in (in the sense that they have classical continuous derivatives up to order ) such that , the moments for , and the collection and is an orthonormal basis of .

Let be an anisotropic vector. For , define the -anisotropy map asIn [35, 36], Triebel has considered anisotropic multiresolution analysis; consider, for any , the closed subspace of spanned by the orthonormal basis , whereThe sequence is an anisotropic multiresolution analysis of in the sense that

(i)

(ii)

(iii)

For , let be the set of pairs where such that at least one component is and whereandClearly the cardinality of is bounded independently of , more preciselyThe following proposition is given in [35, 36] in the case where is the canonical basis of . It remains valid in the case where is any orthonormal basis of .

Proposition 11. *Let be an orthonormal basis of . Let be the coordinates of in . SetThe collection of the union of for and for , , and is then an orthonormal basis of . Thus any function can be written aswithand*

A straightforward extension is given by the following result.

Proposition 12. *The collection of for , , and is an orthonormal basis of . Thus any function can be written as*

##### 3.1. Wavelet Characterization of the Spaces

Let be an anisotropic vector. One of the fundamental properties of the anisotropic wavelet bases is that it characterizes spaces. The results are reminiscent of [5] where the authors have considered the isotropic case .* Without any loss of generality, we will present the results in the canonical basis and drop the letter *. The proofs adapt all steps and arguments of [5] to the anisotropic setting, using(i)the properties of the homogeneous norm ,(ii)anisotropic versions of mean value theorem and Taylor’s theorem with remainder (see [40, 41]),(iii)anisotropic Triebel wavelet characterization of anisotropic Besov spaces (see [35, 36]).

Proposition 13 (the -mean value theorem). *Let be an anisotropic vector. There exist two positive constants and such that for all functions of class on and all ,We denote by the additive subsemigroup of generated by and . In other words, is the set of all numbers as ranges over .*

Proposition 14 (the -Taylor inequality). *Let be an anisotropic vector. Put . Suppose , , and . There are two constants and such that for all functions of class on and all ,where is the so-called -Taylor polynomial of at of -homogeneous degree , given by*

Proposition 15. *Let be an anisotropic vector. Let and . Then if and only if its normis finite, with the usual modification if and/or . This characterization does not depend on the chosen (smooth enough) wavelets and .*

In [35] (section 5.3), a transference method has been proposed with the aim of showing that isotropic and anisotropic Besov spaces are isomorphic together and with the space of all union of sequences and such thatwith the usual modification if and/or . In particular, any embedding theorem for isotropic Besov spaces can be carried over to anisotropic Besov spaces.

*Definition 16. *Let be an anisotropic vector. Let be real number and let and be positive real numbers. One says that a function belongs to if there exists such thatwhere

If then this space was already introduced by Meyer and Xu [43] in order to study local oscillating behaviors.

Proposition 17. *Let be an anisotropic vector. Let , where is the regularity of the Daubechies wavelets. Let and .*(i)*If , then .*(ii)*If , then there exist and such that*

Let and be a function which belongs locally to . For small enough letand

Proposition 18. *Let be an anisotropic vector. Let and be a function which belongs locally to . Then*(1)* is positive and independent of ,*(2)*we have always*(3)*if for some then*

Both Theorem 9 and Proposition 18 yield the following characterization of the pointwise directional exponent.

Corollary 19 (let ). *Let be a unit vector and be any orthonormal basis starting with the vector . Let be the set of all anisotropy vectors such that .**Let . If for some then the exponent of at in direction is given by*

##### 3.2. Wavelet Leaders Characterization

Without any loss of generality, we will present the results in the canonical basis. Let be an anisotropic vector. By we denote a -dyadic rectangle in of scale , which has the formwhere was defined in (66).

Denote by all -dyadic rectangles in of scale .

For setwhere the maximum is taken over all indices giving the same at scale .

Let be a function on . Define its wavelet scaling function asBy Proposition 15, this scaling function does not depend on the chosen (smooth enough) wavelets, andFor such that , the wavelet leader of at a rectangle is defined aswhere is the set formed by the rectangle and all its adjacent rectangles at scale .

*Remark 20. *Note that if then and if then (see [44]).

If , denote by the unique rectangle at the scale that contains .

As in [45], the third point in Proposition 18 yields the following result.

Proposition 21. *Let be an anisotropic vector. Let . If then*

Both Theorem 9 and Proposition 21 yield the following characterization of the pointwise directional exponent.

Corollary 22 (let ). *Let be a unit vector and any orthonormal basis starting with the vector . Let be the set of all anisotropy vectors such that .**Let . If then the exponent of at in direction is given by*

#### 4. Alternative Upper Bound of the Directional Spectra

In Corollary 10, if is an anisotropic vector different from , then the sets are anisotropic but is isotropic. Computing is a very difficult task (see the example of Sierpinski carpet in [46] pages 118–119). To overcome this problem, in [47, 48], we adapted the notion of Hausdorff dimension to the anisotropy ; if , define its -diameter to be . By replacing in the definition of Hausdorff measure, the usual notion of diameter by the -diameter, we easily check (see [49]) that we get the following notion of anisotropic dimension.

*Definition 23. *Let , and let be the set of all coverings of by sets