Wavelets Convergence and Unconditional Bases for 𝐿^𝑝(ℝ^𝑛) and 𝐻¹(ℝ^𝑛)

Abstract

We prove that reasonable nice wavelets form unconditional bases in function space other than . Moreover, characterization of convergence of wavelets series in space and Hardy space has been obtained. Here, is a Banach space with boundedness of Riesz transform.

1. Introduction

Let be a separable Banach space and a sequence of vectors in . We say that is an unconditional basis for if for every there exists a unique sequence of scalars such that where the series converges in the norm and as the net of sums over finite subsets of . In other words we require that converges for every permutation . Again, a consequence of the uniform boundedness principle is that the coefficient functionals are continuous. If is reflexive, then is an unconditional basis for the dual space .

Wavelet systems provide explicit unconditional bases for many function spaces. For instance, any orthonormal wavelet such that and have a common radial decreasing -majorant also generates an unconditional basis for , , and , although it has less regularity.

It can be shown [1] that if two coefficient spaces are equal (as subsets of ), then their norms are equivalent and corresponding Banach spaces , , and Hardy space are (topologically) isomorphic through an isomorphism of one basis to another.

In this paper we will consider the spaces , , and and try to study that the reasonable nice wavelets form unconditional bases in these function spaces. Also, the convergence of wavelet series in these spaces will be characterized. There are several equivalent definitions of real Hardy space . In the real variable theory, one finds (at least) four major types of characterizations of : integrability of maximal functions, integrability of square functions, integrability of conjugate functions (the Riesz transform or its variants), and atomic decompositions.

Let be a classical Calderón-Zygmund singular integral operator on with smooth kernel and let be the associated maximal singular integral where is the truncation at level : The th Riesz transform, , is the singular integral operator The principal value integral above exists for all if is a compactly supported smooth function and one shows for such functions the estimates: for some positive constant independent of . Then a bounded operator can be defined on by the density argument. A subissue arises when one tries to show that the principal value in (4) exists for almost all in for each function in . Following a well-known principle, one looks for estimates for the maximal Riesz transform and one indeed proves that [2, 3] The following results [4] improve inequality (6).

For there exists a constant such that for each function belonging to some , .

Inequality (7) still holds in the limiting case and the answer is provided by next result.

Given , , and a positive constant , there exists a function in such that

Remark 1. Inequality (8) does not hold even for the Hilbert transform, which is the case .

Stein and Weiss originally defined the real Hardy space to be where denotes the Hilbert transform for , and we use Riesz transform.

In order to define real Hardy space , set and and let be the length measure on the segment joining and . For an appropriate constant we have Taking the Fourier transforms on both sides, we obtain Let be a nonnegative continuously differentiable and compactly supported function in the unit ball such that , and set the approximate identity as Convolving identity (10) by approximate identity , we get Set .

Since is a compactly supported function in with zero integral, thus is a function in the Hardy space and so . From (13) we see that Now we define where denotes the Riesz transform.

The Hardy space is initially defined in terms of the atomic decomposition as follows.

Atoms are those measurable functions for which there exist balls in such that for some value of being fixed and denotes the conjugate exponent, .

We say that if has an expansion of the form , and the norm is defined as Atomic Hardy space is a Banach space Notice that , but the converse inclusion is false.

If in the definition of atoms we consider only dyadic intervals in place of balls in , we obtain a “smaller” space called the dyadic real Hardy space . The spaces and are (topologically) isomorphic Banach spaces. This isomorphism leads to a nonconstructive proof of the existence of an unconditional basis for . Namely, the Haar system provides a perfect unconditional basis for , because no regularity is needed in the dyadic setting. In this paper our approach is different and based on the fact that smoother wavelets will directly provide an unconditional basis for ; we will characterize the coefficient space by taking the wavelet basis in .

It should be significant to mention here that it would be better to obtain constants depending only on and regularity of wavelet, but no other information about it. For that reason, a better approach is to prove square function estimates using Calderón-Zygmund decomposition and real interpolation [5].

A Calderón-Zygmund operator is a bounded linear operator with operator norm at most and such that there exists a -function satisfying A function with these properties is called a Calderón-Zygmund kernel or a standard singular kernel.

2. Some Definitions and Auxiliary Results

Definition 2. A Banach space is UMD if for some (and then all, cf. [6]) there is a finite constant so that for all , whenever and , a martingale difference sequence or an arbitrary probability space (i.e., there are sub--algebras ), such that, for all , the function is -measurable and for all .

Remark 3. The validity of Theorem 1.9 of [7] on actually characterizes the UMD-property of . Indeed, let be any complex Banach space and let conclusion of Theorem 1.9 of [7] be satisfied. Since , this is equivalent to the UMD-property of . We also have . This implies that is bounded on .

In view of above remark we can get without modification in [8] the following.

Proposition 4. Let satisfy the standard estimates

Assume, moreover, that is bounded on with operator norm at most . Then is also bounded on , where for all , and it is bounded from to with norm . If, in addition, then is bounded on with norm .

Let us note that atomic definition of is known to agree with the one given in terms of various maximal functions, even for an arbitrary Banach space . One can check that the proof of this fact in the scalar case, as given, for example, in Stein’s book [9], goes through word by word in the general setting for ; the conjugate Hardy space defined as the domain of the Hilbert transform on with the graph norm is always contained in the atomic Hardy space and agrees with it exactly when is a UMD-space. For , we can replace Hilbert transform by Riesz transform and the condition of UMD-space for by the boundedness of Riesz transform in -space.

It is well-known fact that an integral operator satisfying the standard estimates and bounded on is also bounded from to . The square function description of that we have in mind involves the wave expansion of a function in . Recall the definition of the wavelet [10–12]. Let be a set of functions belonging to .

Consider for each , , and . The sequence is a wavelet set if forms an orthonormal basis in . Then is a wavelet basis in and each is a wavelet. The basis is called -regular if and for all , all , and .

The aim of the present paper is to study that reasonable nice wavelets form unconditional bases in function space other than and characterization of convergence of wavelets series in and has been obtained. For this purpose it will be enough to assume that , where is the set of dyadic cube of the form is an orthonormal wavelet basis such that and This is certainly not the weakest possible assumption on for somewhat less restrictive; see [10] (where the class is introduced).

3. Main Results

Theorem 5. Let and let   be a wavelet such that and satisfies condition (26). Then the system is an unconditional basis for and .

Proof. For arbitrary family of , such that , we define on the orthonormal basis by Here is not necessarily unitary, but it is bounded and Now we can write where where is any finite set.
We claim that are Calderón-Zygmund operators with constants uniform in and for that we have to show that satisfy standard estimates uniformly in .
Consider Also for derivatives So we have Condition follows from , which in turn follows from . Therefore from Proposition 4, we know that operators extend continuously to and , and we get inequalities and particularly for , is a finite subset of , , can be written as These inequalities imply that is an unconditional basis for However, it is still not clear that is complete in and .
In particular, if we take , then (34) and imply that
Let us assume that with the product probability measure. First we consider the function of the form for some finite and . Integrating (38) over , interchanging the order of integration, and using Khintchine’s inequality, we get In view of (38), we obtain Similarly, we get Now we take for some finite subset of and define by if and by if , so that . Applying estimate (41) to and using (34), we get By virtue of monotone convergence theorem as finite sets exhaust , we get In order to prove the completeness of in , consider such that with convergence in and there exists a subsequence of partial sums that converge almost everywhere on . By virtue of Fatou’s lemma and (41) we obtain In view of (44) we have Using dominated convergence theorem, we obtain with convergence in . Therefore since is dense in . These facts lead to the conclusion that is complete in . By virtue of (35) this proves that is an unconditional basis for .
Similarly we can prove the result for using (42). Hence the proof is completed.

Theorem 6. Let and let   be a wavelet such that and satisfies condition (26). Then for one has and for one has

Proof. We have where the space is of functions with bounded mean oscillation. is the predual of . We see that defines continuous linear functionals on , , and . If unconditionally in or , then for every by continuity of Now for every we have , with convergence in . Using (41), we obtain Combining (44) and (52), we get (48). Similarly we would handle the case for . Hence the proof is completed.

Corollary 7. For every family of scalars   one has that

Proof. Suppose that is a family of scalars such that . For every we can find finite subset of such that .
Then for every finite subset of , by (44), we have This implies that the series converges unconditionally in to the same -function. Moreover, we have proved that This gives the characterization of the convergence of wavelet series in . Similarly we would characterize convergence in .
Hence the proof is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by University Grants Commission, New Delhi, under the major research Project by F. no. 41-792/2012 (SR).