Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID , 6 pages

http://dx.doi.org/10.1155/2013/365463

Research Article

Relationship between Hardy Spaces Associated with Different Homogeneities and One-Parameter Hardy Spaces

Xinfeng Wu

Department of Mathematics, China University of Mining & Technology, Beijing 100083, China

Received 30 May 2013; Accepted 25 July 2013

Academic Editor: Yongsheng S. Han

Copyright © 2013 Xinfeng Wu. 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.

Abstract.

We prove that the Hardy spaces associated with different homogeneities , are continuously embedded into the intersection of the isotropic Hardy spaces and the nonisotropic Hardy spaces . As a consequence, we obtain that any operator bounded from either or into must be bounded from to .

1. Introduction

For and , we consider two kinds of homogeneities on :

The first are the classical isotropic dilations occurring in the classical Calderón-Zygmund singular integrals, while the second are nonisotropic and related to the heat equations (also the Heisenberg groups). Let and be functions on homogeneous of degree in the isotropic sense and in the nonisotropic sense, respectively, and both smooth away from the origin. Then, it is well known that the Fourier multipliers defined by and given by are both bounded on for , of weak-type , and bounded on the classical isotropic Hardy spaces and, nonisotropic Hardy spaces , respectively. Riviere in [1] asked the question is the composition still of weak-type ? Phong and Stein in [2] answered this question and gave a necessary and sufficient condition for which is of weak-type . The operators Phong and Stein studied are in fact a composition of operators with different kinds of homogeneities which arise naturally in the -Neumann problem. Recently, Han et al. [3] developed a theory of the Hardy spaces , , associated with the different homogeneities and proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on . Weighted function spaces associated with different homogeneities and boundedness of composition of operators on them were recently investigated in [4–6].

The Hardy spaces introduced in [3] have surprising multiparameter structures which reflect the mixed homogeneities arising from the two operators under consideration. A natural question arises: Is there any relationship between and the two classical one-parameter Hardy spaces and ? The main purpose of this paper is to answer this question. We shall prove that are continuously embedded into the intersection of and . As an application, we show that any operator boundedness from either or into must be bounded from into . Our methods are to use the partially discrete Calderón-type formula and the Littlewood-Paley theory in this context, which are appropriately developed.

Before stating the results more precisely, we first recall some notions and notations. For , we denote and . Let satisfy

The isotropic discrete square function is defined by

where , denotes the set of all dyadic cubes with sidelength , and denotes the left-lower corner of . The isotropic Hardy spaces , , are defined by

Similarly, let satisfy

The nonisotropic discrete square function is defined by

where , is the set of dyadic nonisotropic “cubes” with sidelength and , and is the left-lower corner of . The nonisotropic Hardy spaces , , are defined by

For , let . The discrete square function associated with different homogeneities is given by

where denote the set of dyadic rectangles in with sidelength and and is the left-lower corner of . The Hardy spaces , , associated with different homogeneities are defined by

The main result of this paper is as follows.

Theorem 1. Let . One has

More precisely, there is a constant depending on and such that, for all ,

In [3], it was proved that the composition of the two Calderón-Zygmund convolution operators with different homogeneities is bounded on (see [3, Theorem 1.9]). This result can be improved by the following.

Corollary 2. Let . One has that

(1)any operator bounded from either or into must be bounded from to ; (2)if is bounded from to and is bounded from to , then is bounded from to .

2. New Square Function Characterizations of

Let satisfy ,

where is a sufficiently large constant depending on and . Similarly, let satisfy supp ,

Let , , and . Then, can be characterized via the continuous square function defined by

The following Calderón-type identities are well known (cf. [7–9]).

Lemma 3. For , there is a sufficiently large integer such that

where the two series converge in and denotes any fixed point of .

The Calderón-type identities in Lemma 3 lead to the following square functions:

where is the sufficiently large integer in Lemma 3.

The purpose of this section is to prove the following.

Theorem 4. For , one has that

Proof. We only prove that

as can be obtained in the same manner. To prove (18), it suffices to show that

To verify (19), we need the following lemma.

Lemma 5 (see [10]). Let and . Then, for any , , and , one has that

where denotes the classical isotropic Hardy-Littlewood maximal operator and .

We first assume that . Set . Applying the discrete Calderón-type identity in (15), the classical almost orthogonality estimate, and Lemma 5, we deduce that for any , , and ,

Since and are arbitrary points in and , respectively,

which, by the Cauchy-Schwarz inequality, implies that

where we have used . Multiplying from both sides and summing over yield that, for any ,

It follows that

Finally, taking the norm on both sides and applying the Fefferman-Stein vector-valued inequality yield (19). Since is dense in (see [3]), a limiting argument concludes the proof of Theorem 4.

3. Proofs of Theorem 1 and Corollary 2

We first give the following.

Proof of Theorem 1. For , Theorem 1 is trivial since . We now assume that . We only prove as the inequality can be proved similarly. Since is dense in , we may assume that . For , set

Denote that

where is the sufficiently large integer in Lemma 3.

Applying the discrete Calderón-type reproducing formula in Lemma 3,

where and the series converges in the norm. We claim that

Assume the claim for the moment. Then, by the continuous square function characterization of ,

where in the second inequality, we have used Minkowski's inequality, for and in the third inequality used the inequality for .

Thus, to finish the proof of Theorem 1, it suffices to verify claim (29). Since is supported in unit ball of , for , are supported in

Thus, by Hölder's inequality and (which follows from boundedness of ),

We now estimate the last norm by duality argument. For with , applying the Cauchy-Schwarz inequality yields

where . Since

we now have that

which is, in turn, bounded by

where in the first inequality, we have used when . Substituting this estimate back to (32) verifies the claim (29), and hence, Theorem 1 follows.

Finally, we give the following.

Proof of Corollary 2. (1) follows directly from Theorem 1, while (2) follows from (1).

Acknowledgments

This research was supported by NNSF, China (Grant no. 11101423), and supported in part by NNSF, China (Grant no. 11171345). The author would like to express his deep gratitude to the referee for his/her valuable comments and suggestions.

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