Anisotropic Two-Microlocal Spaces and Regularity
We define --anisotropic two-microlocal spaces by decay conditions on anisotropic wavelet coefficients on any --anisotropic wavelet basis of . 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 --two-microlocal spaces are independent of the chosen anisotropic --orthonormal wavelet basis.
Two-microlocal spaces were introduced by Bony  for the study of the propagation of singularities of solutions of hyperbolic PDEs. These spaces became much simpler when Jaffard  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 and the regularity of its fractional derivatives and primitives at . The properties of the two-microlocal domain were investigated by Seuret and Véhel .
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, 6–12]). Anisotropic pointwise Hölder regularity and anisotropic Besov spaces have been introduced (see ). 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  and for the study of semielliptic pseudodifferential operators whose symbols have different degrees of smoothness along different directions see . Two-microlocal spaces have to be changed in order to fit anisotropic behaviors. Let be such that The vector is called anisotropy. For and , we call anisotropic dilation the map In  (resp., ) Calderón and Torchinsky (resp., Folland and Stein) have developed a theory of anisotropic spaces by replacing the Euclidean norm by a homogeneous quasinorm ; recall that is defined on by and, for all , is the unique for which , where is the Euclidean norm on .
The function is continuous and homogeneous in the sense that The corresponding -ball of -radius , centered on , is an ellipse of axis of lengths , centered on .
In the isotropic case ( for all ), the homogeneous quasinorm coincides with the Euclidean norm.
If we set then is also a homogeneous quasinorm in the sense that and there exists a constant such that, for all , The homogeneous quasinorm is equivalent to because In [18–20], we adapted the notion of pointwise regularity to the anisotropy. Let . For , we set and . Thus is the degree of homogeneity of the differential operator , or, as we will say, its homogeneous degree. If , , or is a polynomial, we define its homogeneous degree to be . We also define its -homogeneous degree to be
Definition 1. Let and or let be a function. (1)Let . We say that belongs to (resp., ) if and if there exist a constant , , and a polynomial of -homogeneous degree less than such that respectively, (2)We say that belongs to (resp., ) if and if (8) (resp., (9)) holds for any in with uniform constant .
For , let be a finite set with cardinality bounded independently of , and for let such that there exists such that for all , for all , and for all , Remark that So we write
For , put and . In , the following definition was given.
Definition 2. Let be as above. Let . A homogeneous --anisotropic orthonormal wavelet basis of of order is a family , satisfying the following conditions. (1) (regularity condition).(2)For all , there exists such that, for all , , , and if (localization condition).(3) if (vanishing moment condition).(4) is an orthonormal basis of .
For the inhomogeneous version we need the following definition.
Definition 3. Let be as above. Let . An inhomogeneous --anisotropic orthonormal wavelet basis of of order is a collection of union of two families and , satisfying the following conditions. (1) and (regularity condition).(2)For all , there exists such that, for all and and if (localization condition).(3) if (vanishing moment condition).(4).(5) is an orthonormal basis of .
The inhomogeneous (resp., homogeneous version of) Triebel family of anisotropic wavelets [21, 22] are examples of inhomogeneous (resp., homogeneous) --anisotropic orthonormal wavelet basis of . Let be a one-dimensional multiresolution analysis in with a scaling function (called father wavelet) and an associated wavelet (called mother wavelet) . Let , , be the set of all elements with such that at least one component is an and such that Remark that the cardinality of is bounded, independently of , by .
We easily get the following result.
Proposition 4. Let and Let Suppose that and are Daubechies  (resp., Lemarié-Rieusset and Meyer ) wavelets and let be the common regularity of and resp., . Then the family of resp., the collection of the union of and is a homogeneous (resp., inhomogeneous) --anisotropic orthonormal wavelet basis of of order . We call them the Triebel anisotropic orthonormal bases.
Remark 5. Let be an inhomogeneous --anisotropic orthonormal wavelet basis of of order . For , let and denote the --anisotropic wavelet coefficients of . Then
We now define --anisotropic two-microlocal spaces.
Definition 6. Let and . Let . Define --anisotropic two-microlocal space as the space of functions in such that there exists satisfying
In the next section, we will recall the Mean Value Theorem and Taylor’s theorem with remainder for the homogeneous quasinorm .
In the third section, we will prove the following two theorems which characterize uniform anisotropic regularity (resp., pointwise anisotropic regularity) by decay condition on --anisotropic wavelet coefficients (resp., by --anisotropic two-microlocal spaces); we denote by the additive subsemigroup of generated by and . In other words, is the set of all numbers as ranges over .
Theorem 7. Let . Let be any inhomogeneous --anisotropic orthonormal wavelet basis of , of order , and . Let . (1)If , then there exists a constant such that (2)Conversely, if (19) holds, then if and if .
Theorem 8. Let . Let be any inhomogeneous --anisotropic orthonormal wavelet basis of , of order , and . Let . (1)If , then .(2)Conversely, if and if for , then .(3)If there exist such that , then if and if .
The spaces 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 --anisotropic orthonormal wavelet basis of .
2. Mean Value Theorem and Taylor’s Theorem
The homogeneous quasinorm satisfies the following properties:
The -Mean Value Theorem. There exist two positive constants and such that, for all functions of class on and all ,
The -Taylor Inequality. Suppose , and . There are two constants and such that, for all functions of class on and all , where is the Taylor polynomial of at of homogeneous degree as
Let us first prove Theorem 7.
Proof. (1) Let . We will prove the decay decreasing (19) for the anisotropic wavelet coefficients.
There exists a polynomial of -homogeneous degree less then such that (8) holds for any and in with uniform constant . Since , then the vanishing moment condition for , for implies that
Thanks to the localization condition, we get
But in the sense given in (13). It follows that
(2) Conversely, assume the decay decreasing (19). We will prove that if and if .
We know that is expressed as in (17). Since and , then there exists such that
Then the function is .
Let us study the regularity of .
Since the cardinality of is bounded independently of , then from the localization of it follows that
Let . Denote by the largest value of such that . For , denote by the Taylor polynomial of at of -homogeneous degree (which was defined in (23)):
Then is the -Taylor polynomial of . Its convergence follows from (30) and the fact that if then and .
Let be the unique integer such that . Then
It follows from (29) that
It follows from (30) that
But from the definition of
Take and . Then and . Since is of class , then the -Taylor inequality implies that (i)If , then implies that . From the definition of , we get .(a)If , then (b) If , then Hence We conclude that .(ii)If , then implies that . From the definition of , we get . Therefore We conclude that .
Let us now prove Theorem 8.
Proof. (1) Let and ; then as above
From the localization of the wavelets, we obtain
(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
Similarly we have
As in the first point above, the function is .
Let be the unique integer such that and . As previously
Relation (46) implies that
The assumption for a implies that
So (47) yields
It follows from (35) that
Set and take . Clearly is of class . The -Taylor inequality implies that
As above, if , then
But if , then
We conclude that .
(3) The proof is very similar to the proof of the above point 2 and is left to the reader.
4. Independence of the Wavelet Basis
We will first check that the definition of --anisotropic two-microlocal space does not depend on the chosen --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 --anisotropic orthonormal wavelet basis to another --anisotropic orthonormal wavelet basis is invariant under the action of infinite matrices which belong to an algebra of almost diagonal matrices which was defined in  in order to prove the stability of anisotropic Besov spaces under changes of --anisotropic wavelet bases; therefore, we will then prove that condition (18) is also invariant under this action. Note that is an anisotropic version of the class of almost diagonal matrices that have been considered by Frazier and Jawerth  in the isotropic setting and the corresponding isotropic operator algebras are in the book of Meyer  (resp., Coifman and Meyer [27, 28]).
Definition 9. Let . Denote by the set of infinite matrices indexed by and (where ) and satisfying that where
The following propositions were proved in .
Proposition 10. For all , is an algebra of matrices.
Proposition 11. Assume that . Let .
If and are two homogeneous --anisotropic orthonormal wavelet bases of order , then the matrix, whose coefficients are given by , belongs to for .
Let us recall the following definition .
Definition 12. For every , the algebra is the space of bounded operators on whose matrices on a homogeneous -anisotropic wavelet basis belong to .
Proposition 11 implies that this definition does not depend on the chosen homogeneous --anisotropic wavelet basis of order , for , with .
Theorem 13. Let . If with and , then for all .
Anisotropic two-microlocal spaces with are independent of the chosen --anisotropic wavelet basis.
Proof. We will need the following version of Schur Lemma.
Lemma 14. Let be an infinite matrix and . If then is bounded on , with norm bounded by 1.
Lemma 15. If , then If , , and , then
Proof. We split the sum in the left term in (63) in two sums. (i)If From Lemma 14, if , then , with , and since the cardinality of is bounded independently of , then if because .(ii)If , then
Hence (63) holds.
Let us now prove (64). Since , then