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A Remark on Wavelet Bases in Weighted Spaces
The paper deals with unconditional wavelet bases in weighted spaces. Inhomogeneous wavelets of Daubechies type are considered. Necessary and sufficient conditions for weights are found for which the wavelet system is an unconditional basis in weighted spaces in dependence on .
Wavelet systems in weighted spaces were investigated by several authors. Lemarié-Rieusset considered one-dimensional case, compare . He proved that the homogeneous wavelet system of Daubechies type is an unconditional basis in , , if and only if , where is a weight belonging to the Muckenhoupt class . He also found a sufficient and necessary condition for inhomogeneous systems to be unconditional bases. Other one-dimensional systems were investigated by Kazarian  and García-Cuerva and Kazarian . Multidimensional homogeneous wavelet systems were considered by Aimar et al., compare . They proved that a homogeneous wavelet system satisfying certain regularity conditions is an unconditional basis in if and only if with .
The aim of the paper is to prove the counterpart of Lemarié-Rieusset result for multidimensional inhomogeneous wavelet systems. To formulate the necessary and sufficient condition we use the class of local Muckenhoupt weights introduced by Rychkov in 2001, compare . In his work Rychkov developed the theory of weighted Besov and Triebel-Lizorkin spaces with local Muckenhoupt weight. Wavelets characterizations of the function spaces with so-called admissible weights was given by Haroske and Triebel in . Later Haroske and Skrzypczak proved the characterization for the spaces with Muckenhoupt weights, compare . Recently Izuki and Sawano have proved the result for the function spaces with weights from class , compare , and another approach can be found in .
The main Theorem of the paper asserts that the inhomogeneous wavelet system of Daubechies type is an unconditional basis in if and only if with .
2. Classes of Weights
Let be a nonnegative and locally integrable function on . These functions are called weights, and, for measurable set , denotes . Let denote a space of -integrable functions on with respect to the measure .
2.1. Muckenhoupt Weights
Let us recall the definition of Muckenhoupt weights, compare .
Weight , if and if where supremum is taken over all cubes .
As an example we can take or weights with logarithmic part Then
2.2. Local Muckenhoupt Weights
Definition 2.1 (see ). We define the class of weight to consist of all nonnegative locally integrable functions defined on for which and
It follows directly from definitions that and for any , .
Definition 2.2. We say that if for any where is taken over all measurable sets in .
2.3. Properties of Classes
We would like to mention some important properties of classes .
Lemma 2.4 (see ). Let . Then .
Conversely, if , then for some .
The last lemma implies that . In consequence we can define for a positive number Next lemma shows us an important relation between and weights.
Lemma 2.5 (see ). Let , and be a unit cube, that is, . Then there exists a , such that on and where constant is independent of .
Definition 2.6. Let be locally integrable. Operator where supremum is taken over all cubes in for which , is called local maximal function.
The Fefferman-Stein maximal inequality holds for the operator and local Muckenhoupt weights.
Lemma 2.7 (see ). Let , , and . Then for any sequence of measurable functions , we have
Lemma 2.8 (see ). Let and . Then
It follows from the above lemma that classes are independent of the upper bound for the cube size used in their definition; that is, for any , we could have replaced by in Definition 2.1.
3. Weighted Function Spaces
Following Rychkov we define Triebel-Lizorkin spaces with local Muckenhoupt weights, compare . Because the class of tempered distributions is too narrow for this purpose we introduce a class , which is a topological dual to the following space: where the seminorms are given by We can identify the class with the set of these distributions for which the estimate is valid with some constants depend on . Such a distribution can be extended to a continuous functional on .
We take a function such that and for some . We put and for . Then if . We will write if no vanishing moment holds.
Definition 3.1 (see ). Let , , , and . Let function satisfies where . We define the weighted Triebel-Lizorkin space to be the set of all for which the following quasinorm: is finite.
Remark 3.2. The definition of the above spaces is independent of choice of the function , up to the equivalence of quasinorms. The spaces are quasi-Banach, Banach spaces if and .
4. Characterization by Wavelets
We are going to deal with Daubechies wavelets on . Let be a Daubechies scaling function and a Daubechies wavelet with , , , and . We extend these wavelets from to by the usual tensor product procedure where , , , and and for , where indicates that at least one must be an . The family is an orthonormal basis in , compare .
Definition 4.1. Let , , , and . Then is the collection of all sequences such that
For let us define
The following theorem was obtained in .
Theorem 4.2. Let , , and . For wavelets defined in (4.1), we take
Let . Then if and only if it can be represented as
where and the series converges in . This representation is unique with
is a linear isomorphism of onto .
If , then the system is an unconditional basis in .
5. Spaces with Local Muckenhoupt Weights
In this section we prove our main result. We use the fact that wavelet projection operators satisfy condition (5.2) below. Furthermore we rely on the wavelet characterizations according to Theorem 4.2. We follow the main idea of Aimar et al. . On the other side, we have the wavelet characterization theorem stated in .
Lemma 5.1. Let be a continuous function absolutely bounded by an radial decreasing function such that for all . Then satisfies for some positive real numbers and .
A proof of the above lemma can be found in . Following  we can find that lemma applies to with , where is a Daubechies scaling function. Now for a family , we obtain that it satisfies conditions for every and a positive constant , where is a decreasing sequence of positive real numbers and as .
Theorem 5.2. Let and be a positive Borel measure on finite on compact sets. Let . There exists an unconditional Daubechies wavelet basis in with smoothness if and only if with .
Proof. Let . From Theorem 4.2, we have an unconditional basis in . In  Rychkov shows a Littlewood-Paley characterization of spaces with local Muckenhoupt weights, and it means that . Hence we have an unconditional basis in .
On the other side, Let be a Daubechies wavelet system, which is an unconditional basis in . So every has the representation Operators are uniformly bounded on . We can write , where and if , because wavelets have compact supports, and we deal with locally finite sums. Hence kernels are bounded.
On the other hand, by the properties of the multiresolution analysis, the kernel coincides with if it is regarded as a kernel of projection in . Again, because wavelets and the scaling function are compactly supported, we deal with locally finite sums and get almost everywhere. So the last equality is valid also for spaces, and the kernels are bounded.
We are going to show that is absolutely continuous. Let be a set such that . For every , there exists an open set such that and . Set can be decomposed into a countable union of disjoint and dyadic cubes . From Lemma 5.1, we get that satisfies the same conditions as weakly positive family. Let be a sequence connected with that family. Without, being generality, we assume that , where denote a diameter of and maximum is taken over all cubes from decomposition of . For fixed let be the integer such that . If , we get and . So for every , we have Therefore, , for some constant depending only on and . From weak type inequality for operators , we get Summing over , we have for every . Hence . From Radon-Nikodym Theorem, we get that there exists locally integrable function such that .
Now we can show that . We pick a sequence . Let be a cube with . We can find with . Inequalities where , holds for every . Since operators are of weak type , we get Multiplying both sides by and choosing close to zero, we get for every , . From Lemma 1.4 in , we know that classes are independent of the upper bound for the cube size in their definition. So we get a condition for .
Following , we can state square-function characterization. Let us define where have nonzero integral and and , .
Corollary 5.3. Let and . The following equivalence holds: if and only if .
Proof. Let . From  we have that with norm equivalence Conversely, if we assume that , then from Theorem 4.2 we get that in there exists an unconditional basis. Hence also in , we have an unconditional basis. So from Theorem 5.2, we obtain that .
Remark 5.4. It is known that above statements are not true for general Muckenhoupt weights. Taking for example weight, for , For we have and ; for we have and . Taking big enough, we get that is in , but not in .
Corollary 5.5. Let and . The following equivalence holds: if and only if .
Please note that it follows from the last corollary that if and , then the weighted local Hardy space coincides with if and only if .
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Copyright © 2012 Agnieszka Wojciechowska. 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.