Journal of Function Spaces
Volume 2019 (2019), Article ID 6821464, 10 pages
https://doi.org/10.1155/2019/6821464

Research Article

Strongly Singular Convolution Operators on Herz-Type Hardy Spaces with Variable Exponent

Hongbin Wang¹ and Dunyan Yan²

¹School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China
²School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Correspondence should be addressed to Hongbin Wang; wanghb@sdut.edu.cn

Received 13 May 2019; Accepted 25 June 2019; Published 8 August 2019

Academic Editor: Yongsheng S. Han

Copyright © 2019 Hongbin Wang and Dunyan Yan. 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 investigate the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent.

1. Introduction

The theory of function spaces with variable exponents has been extensively studied by researchers since the work of Kováik and Rákosník [1] appeared in 1991. In [2, 3] the authors defined the Herz-type Hardy spaces with variable exponent and gave some characterizations for them. In [4–7], the authors proved the boundedness of some integral operators on variable function spaces.

Given an open set and a measurable function , denotes the set of measurable functions defined on such that

holds for some .

The set is a Banach function space when it is equipped with the Luxemburg-Nakano norm as follows:

The space is regarded as the variable space, since it generalized the standard space: if is constant, then is isometrically isomorphic to .

The space is defined by

Define to be the set of such that

Define to be the set of such that

and

Denote

Let . The Hardy-Littlewood maximal operator is defined by

where . Let be the set of such that the Hardy-Littlewood maximal operator is bounded on .

Lemma 1 (see [8]). If and satisfies

and

then , that is, the Hardy-Littlewood maximal operator is bounded on .

In addition, we denote the Lebesgue measure and the characteristic function of a measurable set by and , respectively. The notation means that there exist two constants such that .

Next we recall the definition of the Herz spaces with variable exponent. Let and for . Denote and as the sets of all positive and nonnegative integers, respectively, for , if , and .

Definition 2 (see [9]). Let , and . The homogeneous Herz space with variable exponent is defined by

where

The nonhomogeneous Herz space with variable exponent is defined by

where

In [2], the authors gave the definition of the Herz-type Hardy space with variable exponent and the atomic decomposition characterizations. denotes the space of Schwartz functions, and denotes the dual space of . Let be the grand maximal function of defined by

where

and ; is the nontangential maximal operator defined by

with .

Definition 3 (see [2]). Let , and .
(i) The homogeneous Herz-type Hardy space with variable exponent is defined by

and

(ii) The nonhomogeneous Herz-type Hardy space with variable exponent is defined by

and

For , we denote by the largest integer less than or equal to . is the same as in Lemma 9.

Definition 4 (see [2]). Let , and nonnegative integer .

(i)A function on is said to be a central -atom, if it satisfies

(1)supp (2)(3)

(ii)A function on is said to be a central -atom of restricted type, if it satisfies conditions (2), (3) and

(1′)supp

If for some in Definition 4, then the corresponding central -atom is called a dyadic central -atom.

Lemma 5 (see [2]). Let and . Then (or ) if and only if

where each is a central -atom (or central -atom of restricted type) with support contained in and (or ). Moreover,

where the infimum is taken over all above decomposition of .
Let be a smooth radial cut-off function such that if and if . Define the multipliers

where . The kernel for is very singular. Roughly speaking, it looks like

where . Indeed the cancellation is minimal and if one makes a quick computation for , we have

The study of these operators in the context of spaces was carried out by Hirschman [10] and Wainger [11]. Sharp endpoint estimates were obtained by Fefferman and Stein in [12] via the duality of and BMO. Weighted norm and weak(1,1) estimates were established by Chanillo in [13]. The boundedness of these operators on the weighted Herz-type Hardy spaces was proved by Xiaochun Li and Shanzhen Lu in [14].
Motivated by [2, 14], we will study the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent. The main results are as follows.

Theorem 6. Suppose that satisfies conditions (8) and (9) in Lemma 1 and . Then we have

where is independent of .

Theorem 7. Suppose that satisfies conditions (8) and (9) in Lemma 1 and . Then we have

where is independent of .

2. Preliminary Lemmas

Referring to the variable space, there are some important lemmas as follows.

Lemma 8 (see [1]). Let . If and , then is integrable on and

where

The above inequality is named generalized Hölder’s inequality with respect to the variable space.

Lemma 9 (see [9]). Let . Then there exists a positive constant such that, for all balls in and all measurable subsets ,

and

hold, where and are constants with .

Throughout this paper is the same as in Lemma 9.

Lemma 10 (see [9]). Suppose . Then there exists a constant such that, for all balls in ,

Lemma 11 (see [15]). Define a variable exponent by for Then we have

for all measurable functions and .

Lemma 12 (see [16]). Let satisfy conditions (8) and (9) in Lemma 1. Then

for every cube (or ball) , where .
A nonnegative locally integrable function on is said to belong to , if there is a constant such that

where ; denotes a cube in with its sides parallel to the coordinate axes.
The weighted boundedness of has been proved by Chanillo [13].

Lemma 13 (see [13]). Let . Then

Lemma 14 (see [5]). Given a family and an open set , assume that for some and for every ,

Given such that satisfies (8) and (9) in Lemma 1, then for all such that

Since , by Lemmas 13 and 14 it is easy to get the -boundedness of the strongly singular convolution operators .

To prove our main results, we also need the following lemmas.

Lemma 15 (see [11]). The kernel for the multiplier operator is given by

with . Here and depend only on .

Lemma 16 (see [13]). Let and . Then

3. The Proof of Main Results

Firstly we give the proof of Theorem 6.

Proof of Theorem 6. Let . By Lemma 5, we have

where

the infimum is taken over the above decomposition of , and is a dyadic central -atom with the support . Then we have

We first estimate ; by and the -boundedness of we have

Now we estimate . Let

By Lemma 15 and the Minkowski inequality, we have

To estimate the term , we need the pointwise estimate for .
Let . By generalized Hölder’s inequality we have

Therefore, by , , Lemmas 9 and 10, the Minkowski inequality, and generalized Hölder’s inequality we have

What remains is estimating . Let for some , where is the same as the above. Then it follows that

To estimate the term , we need the pointwise estimate for . Let . Then, by the vanishing moment condition on , we have

From the condition of , , if , it follows that

Note that ; that is, . Since , , and , then by Lemmas 9 and 10 we have

Now to estimate , we split as follows:

where is the same as in Lemma 16 and let satisfy .
Applying the mean value theorem to the term brackets in the integrand of , then for we have the pointwise estimate for as follows:

On the other hand, since , by the Minkowski inequality we get

For , using , the pointwise estimate for , and Lemmas 9 and 10 we have

Finally, we estimate . Noting that , we get

Noting , we denote and .
When and , by Lemma 12 we have

When we have

So we obtain

In similar method we can obtain

and

Thus by Lemmas 11, 12, and 16 we have

So by using , , , , and Lemmas 9 and 10 we have

Therefore, by (43), (44), (46), (48), (49), (52), (55), (56), and (64) we complete the proof of Theorem 6.

Similar to the method of Theorem 6, next we give the proof of Theorem 7.

Proof of Theorem 7. Let . By Lemma 5, we have

where

the infimum is taken over the above decomposition of , and is a dyadic central -atom of restricted type with the support . Then we have

We first estimate ; by and the -boundedness of and we have

To estimate , we need the pointwise estimate for .
Suppose that satisfy . Let and , where and . By the vanishing moment condition on , it is easy to prove that . So we have

For , by Lemma 9, the generalized Hölder inequality, and the mean value theorem, we obtain

where .
For , by Lemma 15 we have

Noting that , then . Since for , we obtain

So we have .
For , by the pointwise estimate for in the proof of Theorem 6, we obtain

Using the mean value theorem, we get

where .
For , noting that , we get

Thus, for , we get

So by using , , and Lemmas 9 and 10 we have

Therefore, by (67), (68), and (77) we complete the proof of Theorem 7.

Data Availability

No data were used to support this study.

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 Shandong Provincial Natural Science Foundation, China (Grant No. ZR2017MA041), Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J18KA225), China Postdoctoral Science Foundation (Grant No. 2016M601105), and National Natural Science Foundation of China (Grant Nos. 11761026, 11471309, and 11561062).

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