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Journal of Function Spaces
Volume 2017, Article ID 4340805, 10 pages
https://doi.org/10.1155/2017/4340805
Research Article

Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

Correspondence should be addressed to Shuangping Tao; nc.ude.unwn@psoat

Received 9 October 2017; Accepted 12 December 2017; Published 31 December 2017

Academic Editor: Henryk Hudzik

Copyright © 2017 Yanqi Yang and Shuangping Tao. 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

Let be the singular integral operator with variable kernel defined by and let be the fractional differentiation operator. Let and be the adjoint of and the pseudoadjoint of , respectively. In this paper, the authors prove that and are bounded, respectively, from Morrey-Herz spaces to the weak Morrey-Herz spaces by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product and the pseudoproduct are also given.

1. Introduction

Denote to be the unit sphere in with normalized Lebesgue measure . The singular integral operator with variable kernel is defined by where satisfies the following conditions:

As we all know, the singular integrals with variable kernel played an important role in the theory of nondivergent elliptic equations with discontinuous coefficients (see [1, 2]). Some properties for various of the singular integrals with variable kernel have been obtained by authors; for example, see [36] and their references. In the Mihlin conditions, Calderón and Zygmund proved the boundedness of on the (see [7]).

Let . For tempered distributions , the fractional differentiation operators defined by ; that is, .

Let be the Riesz potential operator of order defined on the space of tempered distributions modulo polynomials by setting It is easy to see that a locally integrable function if and only if . Strichartz (see [8]) showed that is a space of functions modulo constants which is properly contained in , where .

Denote to be the space of spherical harmonical homogeneous polynomials of degree . Let and be an orthonormal system of . It is well known that , , is a complete orthonormal system in (see [9]). Let us expand the function in spherical harmonics where If , then for any . Let Then , defined in (1), can be written as Let and be the adjoint of and the pseudoadjoint of , respectively, defined by

Let us give some necessary notations. In the following, unless otherwise stated, for a -measurable set , denotes its characteristic function. We use the symbol to denote that there exists a positive constant such that . For any index , we denote by its conjugate index; that is, .

Let and be the operators defined in (1) which are differentiated by their kernels and . Let , denote the product and pseudoproduct of and , respectively. In [7], Calderón and Zygmund found that these operators are closely related to the second order linear elliptic equations with variable coefficients and established the following boundedness of the operators , , , , and on .

Theorem A (see [7]). Let , , satisfy (2) and (3). Then(1);(2);(3)

In 2015, Chen and Zhu proved that Theorem A was also true on Weighted Lebesgue space and Morrey space (see [10]). In 2016, Tao and Yang obtained the boundedness of those operators on the weighted Morrey-Herz spaces (see [6]). A natural question is whether these operators also have boundedness on the weak Morrey-Herz spaces. The answer is affirmative. The main purpose of this paper is to generalize the above results to the cases of weak Morrey-Herz spaces (see Definition 7 in the next section).

Our main results are stated as follows.

Theorem 1. Let , and . Assume that is defined by (1) and , which satisfies (2) and (3), meets the following condition: Then one has(1);(2).

Theorem 2. Let , , and . Suppose that and satisfy (2) and (3). If satisfies (9) and satisfies then one has

Furthermore, we also consider the cases and . As we all know, is the square root of Laplacian operator and is the identity operator . In this case, we obtain the following results.

Theorem 3. Let and . Suppose that satisfies (2), (3), and (10). Then one has(1);(2);(3).

Theorem 4. Let and . Suppose that satisfies (2), (3), and Then one has(1);(2).

Theorem 5. Let and . Suppose that and satisfy (2) and (3). If satisfies (10) and satisfies (12), then one has

2. Preliminaries and Main Lemmas

In this section, we shall recall the definitions of the homogeneous Morrey-Herz spaces and weak Morrey-Herz spaces. Furthermore, the weak estimates of defined by (6) and a class of Calderón-Zygmund operators will be established on Morrey-Herz spaces.

The well-known Morrey spaces, introduced originally by Morrey [11] in relation to the study of partial differential equations, were widely investigated during last decades, including the study of classical operators of harmonic analysis in various generalizations of these spaces. Morrey-type spaces appeared to be quite useful in the study of the local behavior of the solutions of partial differential equations, a priori estimates, and other topics. They are also widely used in applications to regularity properties of solutions to PDE including the study of Navier-Stokes equations (see [12] and references therein). The ideas of Morrey [11] were further developed by Campanato [13]. A more systematic study of these (and even more general) spaces, we refer the readers to see [12, 1421]. In 1964, Beurling [22] first introduced some fundamental forms of Herz spaces to study convolution algebras. Later Herz [23] gave versions of the spaces defined below in a slightly different setting. Since then, the theory of Herz spaces has been significantly developed, and these spaces have turned out to be quite useful in harmonic analysis. For instance, they were used by Baernstein and Sawyer [24] to characterize the multipliers on the classical Hardy spaces and used by Lu and Yang [25] in the study of partial differential equations. More results and further details can be found in [2628]. On the basis of above available results, the theory of the homogeneous Morrey-Herz spaces goes back to Lu-Xu [29] who considered the boundedness of a class of sublinear operators; also see [6, 30, 31] for more further results.

Next we give the following notation. For each , we denote and , .

Definition 6 (see [29]). Let , , , and . The homogeneous Morrey-Herz spaces are defined by where and the usual modifications should be made when .

In what follows, for any and , let

Definition 7 (see [29]). Let , , , and . A measurable function is said to belong to the homogeneous weak Morrey-Herz spaces , if where where the usual modifications are made when .

Lemma 8 (see [32]). is said to be a homogeneous space. Let , , , , , and , if a sublinear operator meets the following requirements: And is bounded from to , and then is a bounded operator from to . When , as can be seen from the proof process, the above conclusion is still valid.

Lemma 9. Let be defined by (6). Then is weak , and

Proof. Let and ([7]). Fix and . Form the Calderón-Zygmund decomposition of at height . Then for any , there exists a decomposition of such that(1),(2), where are finite overlapping, and We now decompose as the sum of two functions, and , defined by where Then almost everywhere, and is supported on and has zero integral. Since , We estimate the first term, in view of the fact that ([10]). Without loss of generality, for , we have Let be the cube with the same center as and whose sides are twice as long, and let . Then we have , and together with characteristic of left in (2), we can obtain Thus we have Notice that almost everywhere. Hence, to complete the proof of the weak (1, 1) inequality it will suffice to show that For any , , the formula is still valid. Denote the center of by , and then we have Noticing that it follows thatThen for any , it is clear that Summing up the estimates above for and , we finish the proof of Lemma 9.

Lemma 10. Let , , and . Let be a generalized Calderón-Zygmund operator, and then is bounded from to ; namely,

Proof. It is well known that is weak (1) (e.g., see [33]). Noticing that satisfying (18) with , then we can obtain Lemma 10 by using Lemma 8.

Lemma 11. Let , , and ; defined by (6) is bounded from to , and

Proof. By applying the fact that and ([7]), we can easily obtain Noticing that the interior integral above is meeting the condition in (18) for and is weak (1) on the basis of Lemma 9, it is not difficult to deduce that is bounded from to . Hence This completes the proof of Lemma 11.

Lemma 12. Let be a homogeneous of degree and locally integrable in . Let and If and , , then, for , , and , one has

Proof. Let . For all with , then satisfies the following inequalities: This, together with the boundedness of on (see [34]), tells us that is a generalized Calderón-Zygmund operator ([33]). Thus, by applying Lemma 10, we see that is bounded from to with bound for , , and .
Therefore, the proof of Lemma 12 is finished.

Lemma 13. Let and be a singular operator which is defined by where , , and , for , , and then one has that, for , , , and , the operator

Proof. With an argument similar to that used in the proof of Lemma  5.2 in [10], it is not difficult to obtain Lemma 13. Thus, we omit the details here.

3. Proofs of Theorems

Proof of Theorem 1. Let From [3], for any , we can write the coefficients aswhere
We will firstly prove conclusion . Write By (43), it follows that Then, we have by (9)Moreover, by the fact that is a generalized Calderón-Zygmund operator (see [35]), which is defined by Thus we can get that is bounded from to by applying Lemma 10; namely, Then by (see [36]), (46), (48), and Lemma 11, we have Now let us turn to estimate . By applying the definition of and we can deduce that In order to estimate norm of , we first consider for any fixed . Noting that , for any , then we have Thus, by (48) and Lemma 11, we get Further, we estimate the norm of . From the fact that is a generalized Calderón-Zygmund operator with kernel (see [10]) then we get by Lemma 10Then, combining (52) with (54), we have By estimates (46), (50), and (55), we get Thus we finish the proof of Theorem 1.

Proof of Theorem 2. Let Write where For any , with a similar argument used in the proof of Theorem 1 in terms of (9) and (10), we can obtain that Let and Since and satisfy (3), then we get Write ([10])Then Therefore, together with (55), (60), and Lemma 11, we obtain This finishes the proof of Theorem 2.

Proof of Theorem 3. We can estimate term (1) exactly as we did for the corresponding boundedness in Theorem 1 in the above arguments. Thus, we have only to prove (2) and (3) of Theorem 3. In order to do this, we use the same notations as in the proof of Theorem 2. By using the fact that and satisfy (10), therefore, we have Firstly, let us prove (2). As in the proof of Theorem 1, we can get As is a special Calderón-Zygmund operator, it is bounded from Morrey-Herz spaces to weak Morrey-Herz spaces by applying Lemma 10. Thus we have Then by (65), we get