Journal of Function Spaces
Volume 2018 (2018), Article ID 1296837, 10 pages
https://doi.org/10.1155/2018/1296837

Research Article

Herz-Type Hardy Spaces Associated with Operators

Yan Chai, Yaoyao Han, and Kai Zhao

School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

Correspondence should be addressed to Kai Zhao; zhkzhc@aliyun.com

Received 14 May 2018; Revised 28 June 2018; Accepted 8 July 2018; Published 17 July 2018

Academic Editor: Alberto Fiorenza

Copyright © 2018 Yan Chai et al. 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.

Suppose is a nonnegative, self-adjoint differential operator. In this paper, we introduce the Herz-type Hardy spaces associated with operator . Then, similar to the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we prove the atomic and molecular decompositions of the Herz-type Hardy spaces associated with operator . As applications, the boundedness of some singular integral operators on Herz-type Hardy spaces associated with operators is obtained.

1. Introduction

As we know, the theory of function spaces constitutes an important part of harmonic analysis and partial differential equations. Some results of the classical Hardy spaces can be found in [1–6], etc. Since there are some important situations in which the theory of classical Hardy spaces is not applicable, many authors begin to study Hardy spaces that are adapted to the differential operator . For example, Auscher, Duong, and McIntosh [7], then Duong and Yan [8, 9], introduced the Hardy and BMO spaces adapted to the operator which satisfies the Gaussian heat kernel upper bounds. Yang and his cooperators discussed new Orlicz-Hardy spaces associated with operators [10–13]. For more results, we refer to [14–19] and the references therein.

It is known that many classical function spaces and the Hardy type spaces associated with operators have the atomic decompositions and the molecular decompositions, and the atomic and molecular decompositions of function spaces make the linear operators acting on spaces very simple; see [20–29], etc. In fact, the characterizations of spaces of functions or distributions, including the atomic and molecular characterizations, have many important applications in harmonic analysis. In recent years, it has been proved that many results in the classical theory of Hardy spaces and singular integrals can transplant to the function spaces associated with operators, such as [30–36].

Suppose is a nonnegative, self-adjoint differential operator, and has -calculus on . The kernel of satisfies the Gaussian upper bound on . Motivated by [17, 20, 37], etc, in this paper, we use the area integral function associated with the operator to define the Herz-type Hardy space . In order to obtain the atomic and molecular decompositions of the Herz-type Hardy space, the -atom and the -molecule are introduced. By the method of the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we characterize spaces for atoms and molecules; that is, we prove the atomic and molecular decompositions of Herz-type Hardy spaces associated with the operator . Finally, as applications, we prove some singular integral operators are bounded from to Herz spaces and also bounded on .

Throughout the paper, we always use the letter to denote a positive constant, which may change from one to another and only depends on main parameters. We also use to denote the characteristic function of which is the subset of .

2. Preliminaries

For convenience, we recall the definitions of Herz and Herz-type Hardy spaces on . For details, we refer to [37, 38], etc.

Let , , and .

Definition 1 (see [38]). Let , , .
The homogeneous Herz space is defined by

where

The nonhomogeneous Herz space is defined by

where

Let be the grand maximal function of defined as , where , .

Definition 2 (see [38]). Let .
The homogeneous Herz-Hardy space is defined by

Moreover,

The nonhomogeneous Herz-Hardy space is defined by

Moreover,

Now, we introduce the Herz-type Hardy spaces associated with operators.
Suppose that the differential operator satisfies the following two assumptions.

Assumption (A1). is a nonnegative self-adjoint operator on , and has a bounded -functional calculus in .

Assumption (A2). Each of the heat semigroup generated by has the kernel which satisfies the following Gaussian upper bounds; i.e., there exist constants such that

Obviously, the typical second-order elliptic or subelliptic differential operators are to satisfy these assumptions (see for instance, [39]).

Now we introduce the following lemmas which will be used in this paper.

Lemma 3 (see [40, 41]). Let be a nonnegative self-adjoint operator satisfying Assumptions (A1) and (A2). For every , there exist two positive constants such that the kernel of the operator satisfies

for all and almost every .

Lemma 4 (see [19]). Let be even and . Suppose denotes the Fourier transform of . Then for each , kernel of satisfies

and

for all and

Lemma 5 (see [19]). For , we define

Then for any nonzero function , .

Lemma 6 (see [19]). Let . Then, for any ,

Definition 7 (see [8]). For any , the area integral function associated with operators is defined by

where , and satisfies Assumptions (A1) and (A2).

Thus, for any integer , , the kernel of satisfies , where is a positive decreasing function, satisfying , for any . Therefore, for convenience, in the following, we always set in (15).

By the definition (15) and Assumptions (A1) and (A2), it is easy to check that for any , ; there exist such that (or see, for example, [11]):

Definition 8. Suppose . Let satisfy Assumptions (A1) and (A2).
The homogeneous Herz-type Hardy space associated with the operator is defined by

The norm of in is

The nonhomogeneous Herz-type Hardy space associated with the operator is defined by

The norm of in is

Since , the Herz-type Hardy spaces associated with the operator introduced in Definition 8 are really the expansion of Hardy space associated with operators in [8].

3. The Decompositions of Herz-Type Hardy Spaces

In this section, we give the atomic decomposition and molecular decomposition of Herz-type Hardy spaces associated with the operator , respectively, which are the main results in this paper.

3.1. Atomic Decomposition of the Herz-Type Hardy Space

For the purpose of the atomic decomposition of the Herz-type Hardy space associated with operator, we first introduce the -atom.

Definition 9. Let , , . Set , where satisfies Assumptions (A1) and (A2).
A function is said to be an -atom, if there exists , such that

(i);(ii);(iii);

, .
Function is said to be a restrictive -atom, if there exists , satisfying (i), (ii), (iii), and

The main result of this subsection is the following atomic decomposition of the Herz-type Hardy spaces associated with the operator . The part of the idea is from [1].

Theorem 10. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of -atoms and a sequence of numbers such that can be represented in the following form:

and the sum converges in the sense of -norm, . Moreover,

where the infimum is taken over all of the decompositions of .

Proof. First, we prove the theorem for .
Necessity. Let and be the same as those in Lemma 4. Set . Then by -functional calculus (see for example, [42]), for every , there is

Set , . denotes the collection of all dyadic cubes in . Let . Then, for any , there exists only one such that . Let ; i.e., denote the collection of maximal dyadic cubes in . Set

where is the side length of .
Then, by (23), we have that

where and

We will prove that, up to a normalization by a multiplicative constant, every is an -atom.
Obviously, by Lemma 4, we conclude that , .
For , and , then, by Hölder inequality together with Lemma 6, we have that

Furthermore, we prove the following estimate:

Noting the definition of in (26) and Definition 8 for , it means that we should establish the following inequality:

where is the characteristic function of .
It is sufficient to show that

In fact, if , then , . Let denote the characteristic function of . Thus, by the definition of , we obtain that

Therefore,

Hence, the necessity is proved.
Sufficiency. Let , where every is an -atom. We will prove the sufficiency for two situations: and .
If , then, to prove , it is only need to show that, for any -atom , there exits a constant independent of such that

In fact, then, we have

Suppose that is an -atom with and for any () , . Then

For , boundedness of and the size condition of atom tell us

In order to estimate , we write

For , noting that and , if , , then . Thus, by Lemma 3, we can have that

For , noting that , if , , then . So that holds true. Therefore, we can obtain that

Hence, combining (38) and (39), one can have

If , then

For , boundedness of and the Hölder inequality tell us

For , similar to the estimate of , we can obtain the estimates of as (38) and (39). Thus, using the Hölder inequality, we have that

The sufficiency is proved. Then the proof of Theorem 10 for is finished.
If , the proof that is exactly similar to the situation of , we only need to slightly modify some formulas above. We should set

Inequalities (29) and (30) are replaced with

and

respectively. To obtain inequality (46), there is

Other details are omitted.

For the nonhomogeneous Herz-type Hardy space associated with the operator , there is the same result as follows.

Theorem 11. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of the restrictive -atoms and a sequence of numbers such that can be represented in the following form:

and the sum converges in the sense of -norm, . Moreover,

where the infimum is taken over all of the decompositions of .

3.2. Molecular Decomposition of the Herz-Type Hardy Space

In this subsection, we first introduce -molecule in the following; then we give the molecular decomposition of the Herz-type Hardy space associated with operator.

Definition 12. Let , , . Set , where satisfies Assumptions (A1) and (A2).
A function is said to be an -molecule, if there exists , such that

(i);(ii) and ,

where , ; and

Function is said to be a restrictive -molecule, if there exists , satisfying (i), (ii), and

It is not difficult to check that an -atom associated with ball is also an -molecule associated with the same ball .

The following molecular decomposition of the Herz-type Hardy spaces associated with the operator is the main result in this subsection.

Theorem 13. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of -molecules and a sequence of numbers such that can be represented in the following form:

and the sum converges in the sense of -norm, . Moreover,

where the infimum is taken over all of the decompositions of .

Proof. (i) The proof of necessity is a direct consequence of the necessity in Theorem 10, since an -atom is also an -molecule for all .
(ii) The proof of sufficiency is similar to that of the sufficiency in Theorem 10. The main difference is that the support of -molecule is not the ball . However, we can overcome this difficulty by decomposing into annuli associated with the ball , then using the same argument as in Theorem 10 to get sufficiency. We omit the details here.

Similarly, for the nonhomogeneous Herz-type Hardy space associated with operator , there is the same result as follows.

Theorem 14. Let , , . Suppose satisfies Assumptions (A1) and (A2). Then, if and only if there exist a family of the restrictive -molecules and a sequence of numbers such that can be represented in the following form:

and the sum converges in the sense of -norm, . Moreover,

where the infimum is taken over all of the decompositions of .

4. Boundedness of Singular Integral Operators

This section is based on the decompositions of in the previous section; as applications, we give some boundedness of sublinear operators satisfying certain conditions on Herz-type Hardy spaces associated with operator.

Theorem 15. Let , , . If a sublinear operator satisfies that

(i) is bounded on ;(ii) satisfies the size condition

for suitable function with .
Then is bounded from to , that is,

Proof. Suppose . By Theorem 10, we have

where each is a -atom with , and , , .
Thus,

First, we estimate . By the boundedness of in , we can infer that

Therefore, if , then, by the Jensen inequality, we have

If , let . Then we can obtain

Hence, .
Second, we estimate . By the Hölder inequality and (55), we obtain

Thus,

Therefore, if , by the Jensen inequality, we have

If , by the Hölder inequality, we obtain

Hence,

Theorem 16. Let , , . Suppose that is a sublinear operator as Theorem 15 and and are commutative. Then is bounded on .

Proof. Suppose is an -atom. According to Definition 9, there exists , such that

(i);(ii);(iii)

being an -molecule only needs to be prove, such that

(1);(2) and

In fact, it is enough to check . For any and , we have

Thus, we complete the proof of Theorem 16.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (11471176).

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