Journal of Function Spaces

Journal of Function Spaces / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 6821464 | 10 pages | https://doi.org/10.1155/2019/6821464

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

Academic Editor: Yongsheng S. Han
Received13 May 2019
Accepted25 Jun 2019
Published08 Aug 2019

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 [47], 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 thatholds 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 thatDefine to be the set of such thatandDenote

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

Lemma 1 (see [8]). If and satisfiesand 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 bywhereThe nonhomogeneous Herz space with variable exponent is defined by whereIn [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 bywhereand ; is the nontangential maximal operator defined bywith .

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

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 ifwhere 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 multiplierswhere . The kernel for is very singular. Roughly speaking, it looks likewhere . Indeed the cancellation is minimal and if one makes a quick computation for , we haveThe 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 havewhere 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 andwhere 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 ,andhold, 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 havefor all measurable functions and .

Lemma 12 (see [16]). Let satisfy conditions (8) and (9) in Lemma 1. Thenfor 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 bywith . 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 havewherethe infimum is taken over the above decomposition of , and is a dyadic central -atom with the support . Then we haveWe first estimate ; by and the -boundedness of we haveNow we estimate . LetBy 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 haveWhat remains is estimating . Let for some , where is the same as the above. Then it follows thatTo estimate the term , we need the pointwise estimate for . Let . Then, by the vanishing moment condition on , we haveFrom the condition of , , if , it follows thatNote that ; that is, . Since , , and , then by Lemmas 9 and 10 we haveNow 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 getFor , using , the pointwise estimate for , and Lemmas 9 and 10 we haveFinally, we estimate . Noting that , we get