Anisotropic Two-Microlocal Spaces and Regularity

We define D-u-anisotropic two-microlocal spaces by decay conditions on anisotropic wavelet coefficients on anyD-u-anisotropic wavelet basis of L(R). We prove that these spaces allow the characterizing of pointwise anisotropic Hölder regularity. We also prove an anisotropic wavelet criterion for anisotropic uniform regularity. We finally prove that both this criterion and anisotropic D-u-two-microlocal spaces are independent of the chosen anisotropicD-u-orthonormal wavelet basis.


Introduction
Two-microlocal spaces were introduced by Bony [1] for the study of the propagation of singularities of solutions of hyperbolic PDEs.These spaces became much simpler when Jaffard [2] characterized them by decay conditions on isotropic wavelet coefficients.These spaces yield very accurate information on the local oscillations of a function  near a point  0 and the regularity of its fractional derivatives and primitives at  0 .The properties of the two-microlocal domain were investigated by Seuret and Véhel [3].
However, many natural mathematical objects, as well as many multidimensional signals and images from real physical problems, present strong anisotropies (see [4,5] and references therein).For instance, this is the case in the textures of medical images (mammographies, osteoporosis, muscular tissues, etc.), hydrology, fracture surfaces analysis, and so forth (see [4,[6][7][8][9][10][11][12]).Anisotropic pointwise Hölder regularity and anisotropic Besov spaces have been introduced (see [13]).Anisotropic Besov spaces have played a central role in the mathematical modeling of anisotropic textures.They also have been used to study some PDEs; see [14] and for the study of semielliptic pseudodifferential operators whose symbols have different degrees of smoothness along different directions see [15].Two-microlocal spaces have to be changed in order to fit anisotropic behaviors.Let u = ( 1 , . . .,   ) be such that The vector u is called anisotropy.For  > 0 and  = ( 1 , . . .,   ) ∈ R  , we call anisotropic dilation the map In [16] (resp., [17]) Calderón and Torchinsky (resp., Folland and Stein) have developed a theory of anisotropic H  (R  ) spaces by replacing the Euclidean norm by a homogeneous quasinorm  u ; recall that  u is defined on R  by  u (0) = 0 and, for all  ̸ = 0,  u () is the unique  > 0 for which | −u | = 1, where | ⋅ | is the Euclidean norm on R  .

Journal of Function Spaces
In the isotropic case (  = 1 for all 1 ≤  ≤ ), the homogeneous quasinorm  u coincides with the Euclidean norm.
We now define -u-anisotropic two-microlocal spaces.
Definition 6.Let  > 0 and − ≤   ≤ 0. Let  0 ∈ R  .Define -u-anisotropic two-microlocal space  ,  ,u ( 0 ) as the space of functions  in  2 (R  ) such that there exists  > 0 satisfying In the next section, we will recall the Mean Value Theorem and Taylor's theorem with remainder for the homogeneous quasinorm  u .
In the third section, we will prove the following two theorems which characterize uniform anisotropic regularity (resp., pointwise anisotropic regularity) by decay condition on -u-anisotropic wavelet coefficients (resp., by u-anisotropic two-microlocal spaces); we denote by Δ the additive subsemigroup of R generated by 0, 1,  2 / 1 , . . .and   / 1 .In other words, Δ is the set of all numbers () as  ranges over N  . (2) The spaces  ,  ,u ( 0 ) are defined by conditions on the wavelet coefficients; therefore we should check that this definition is independent of the wavelet basis chosen.This will be done in the fourth section.We will also show that Theorem 7 does not depend on the chosen -u-anisotropic orthonormal wavelet basis of  2 (R  ).

Mean Value Theorem and Taylor's Theorem
The homogeneous quasinorm  u satisfies the following properties: In [16,17], there are versions of Mean Value Theorem and Taylor's theorem with remainder for the homogeneous quasinorm  u .Using the fact that  u and | ⋅ | u are equivalent we deduce the following results.(
There are two constants   > 0 and ] > 0 such that, for all functions  of class  (+1) on R  and all ,  ∈ R  , where  is the Taylor polynomial of  at  of homogeneous degree  as

Proofs of Theorems 7 and 8
Let us first prove Theorem 7.
Let  0 be the unique integer such that From the definition of  u , we get  u () ≥ .
From the localization of the wavelets, we obtain Therefore (2) Conversely, assume that (44) holds; then Since the cardinality of   is bounded independently of , then from the localization of the wavelets it follows that        ()      ≤  (2 Similarly we have As in the first point above, the function is Let  0 be the unique integer such that 2 − 0 ≤ | −  0 | u < 2 ⋅ 2 − 0 and  1 =  0 /.As previously Relation (46) implies that The assumption  ∈   u (R  ) for a  > 0 implies that We conclude that  ∈   u,log ( 0 ).(3) The proof is very similar to the proof of the above point 2 and is left to the reader.

Independence of the Wavelet Basis
We will first check that the definition of -u-anisotropic twomicrolocal space  ,  ,u ( 0 ) does not depend on the chosen -u-anisotropic orthonormal wavelet basis.We will check a stronger (but simpler) requirement which implies that the condition considered has some additional stability; indeed, we will first prove that the matrix of the operator which maps a -u-anisotropic orthonormal wavelet basis to another u-anisotropic orthonormal wavelet basis is invariant under the action of infinite matrices which belong to an algebra M u  −1 , of almost diagonal matrices which was defined in [13] in order to prove the stability of anisotropic Besov spaces under changes of -u-anisotropic wavelet bases; therefore, we will then prove that condition (18) is also invariant under this action.Note that M u  −1 , is an anisotropic version of the class of almost diagonal matrices that have been considered by Frazier and Jawerth [25] in the isotropic setting and the corresponding isotropic operator algebras are in the book of Meyer [26] (resp., Coifman and Meyer [27,28]). where The following propositions were proved in [13].Let us recall the following definition [13].
Definition 12.For every  > (  − 1), the algebra O(M u , ) is the space of bounded operators on  2 (R  ) whose matrices on a homogeneous -anisotropic wavelet basis belong to M u , .
Proposition 11 implies that this definition does not depend on the chosen homogeneous -u-anisotropic wavelet basis of order , for  <  1 max{  ∈ Δ;   < []}, with  ≥ 1.
Proposition 11 and the second point in Lemma 15 below yield the following theorem.Anisotropic two-microlocal spaces  ,  ,u ( 0 ) with − ≤   < 0 are independent of the chosen -u-anisotropic wavelet basis.
Proof.We will need the following version of Schur Lemma.