Toeplitz Type Operators Associated with Generalized Calderón-Zygmund Operator on Weighted Morrey Spaces

Abstract

Let be a generalized Calderón-Zygmund operator or (the identity operator), let and be the linear operators, and let . Denote the Toeplitz type operator by , where and is the fractional integral operator. In this paper, we investigate the boundedness of the operator on weighted Morrey space when belongs to the weighted BMO spaces.

1. Introduction and Results

The classical Morrey spaces, introduced by Morrey [1] in 1938, have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations (see [2, 3]). Komori and Shirai [4] introduced a version of the weighted Morrey space , which is a natural generalization of the weighted Lebesgue space .

Definition 1. Suppose that is a linear operator with kernel defined initially by The operator is called a generalized Calderón-Zygmund operator provided that the following three conditions are satisfied: (1)can be extended into a continuous operator on .(2) is smooth away from the diagonal with where is a constant independent of and .(3)There is a sequence of positive constant numbers such that, for each ,where is a fixed pair of positive numbers with and .

If we compare the generalized Calderón-Zygmund operator with the classical Calderón-Zygmund operator, whose kernel is smooth away from the diagonal with where for some , we can find out that the classical Calderón-Zygmund operator is a generalized Calderón-Zygmund operator defined above with , , and any .

Let be a locally integrable function on . The Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator is defined by where is the generalized Calderón-Zygmund operator or (the identity operator), and are the linear operators, , and .

Note that the commutators are the particular cases of the Toeplitz type operators . The Toeplitz type operators are the nontrivial generalization of these commutators.

The boundedness of the singular integral commutators generated by BMO function was obtained in [5–8]. Motivated by these, in this paper, we investigate the boundedness of on the weighted Morrey space when belongs to weighted BMO space and we have the following result.

Theorem 2. Suppose that is a Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator , and . Let , , , , , and the critical index of for the reverse Hölder condition . If for any , and are the bounded operators on , and then there exists a constant such that

The following results are immediately obtained from Theorem 2.

Corollary 3 (see [5]). Let , , , , and . Suppose that and the critical index of for the reverse Hölder condition ; then, is bounded from to .

Corollary 4. Suppose that is a Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator and . Let , , , , and . If for any , and are the bounded operators on ; then, there exists a constant such that

The paper is organized as follows. In Section 2, we will introduce some notation and definitions and recall some preliminary results. In Section 3, we give the sharp estimates for . In Section 4, we will give the proof of Theorem 2.

2. Some Preliminaries

First, let us recall some notation and the definition of weight classes.

A weight is a nonnegative, locally integrable function on . Let denote the ball with the center and radius , and let for any . For a given weight function and a measurable set , we also denote the Lebesgue measure of by and set weighted measure . For any given weight function on , , denote by the space of all function satisfying

Definition 5 (see [9]). A weight is said to belong to the Muckenhoupt class for , if there exists a constant such that for every ball . The class is defined by replacing the above inequality with When , we defined .

Definition 6 (see [10]). A weight function belongs to for , if for every ball in , there exists a positive constant which is independent of such that where denotes the conjugate exponent of ; that is, .

From the definition of , we can get that

Definition 7 (see [11]). A weight function belongs to the reverse Hölder class if there exist two constants and such that the following reverse Hölder inequality holds for every ball .

It is well known that if with , then for all and for some . If with , then there exists such that . It follows directly from Hölder’s inequality that implies for all Moreover, if , , then we have for some . We write to denote the critical index of for the reverse Hölder condition.

Lemma 8 (see [11]). Suppose . Then, there exist two constants and , such that

Lemma 9 (see [11]). Let , . Then, for any ball and any , there exists an absolute constant such that where does not depend on or .

Next, we will recall the definition of the Hardy-Littlewood maximal operator and several variants, the fractional integral operator, and some function spaces.

Definition 10. The Hardy-Littlewood maximal operator is defined by For , the sharp maximal operator is defined by For , we define the fractional maximal operator by and define the fractional weighted maximal operator by where the above supremum is taken over all balls containing . In order to simplify the notation, we set and .

Definition 11. For , the fractional integral operator is defined by

Lemma 12. Let be fractional integral operator, and let be a measurable set in . Then, for any , there exists a constant such that

Proof. Sincewe have

Definition 13. Let and be a weighted function. A locally integrable function is said to be in if where and the supremum is taken over all balls .

Lemma 14 (see [12]). Let . Then, for any , there exists an absolute constant such that .

Definition 15 (see [4]). Let , , and let be a weight function. Then, the weighted Morrey space is defined by where and the supremum is taken over all balls .

In order to deal with the fractional order case, we need to consider the weighted Morrey space with two weights.

Definition 16 (see [4]). Let and . Then, for two weights and , the weighted Morrey space is defined by where and the supremum is taken over all balls .

We list a series of lemmas which will be used in the proof of our theorem.

Lemma 17 (see [5]). Let , , , , and . Then, for any , we have

Lemma 18 (see [5]). Let , , , , and . Then, for every and , we have

Lemma 19 (see [5]). Let , , , and . Then, if and , we have

Lemma 20 (see [4]). Let , , , and . Then, if, we have

3. The Sharp Estimates for

In this section, we will prove the sharp estimates for as follows.

Theorem 21. Suppose that is a Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator , and . Let , , , , , , and . If for any , then there exists a constant such that

Proof. For any ball which contains , without loss generality, we may assume that is a generalized Calderón-Zygmund operator. We write, by , where Since , then We are going to estimate each term, respectively. Since is bounded from to and , then, by Kolmogorov’s inequality and Lemmas 8 and 9, we get Since , by Hölder’s inequality and Lemma 12, we have By the definition of generalized Calderón-Zygmund operator, we haveThen, by Hölder’s inequality, we get Since and , then there exists such that . Applying Hölder’s inequality for , , and and by (3) of Definition 1, we get Note that then, Since , we have . By , there is such that . Let ; then, By , we have . Then, ; that is, . Then, we haveApplying Hölder’s inequality for and , the definition of weighted BMO, and , we have Hence,Note that Thus, by Hölder’s inequality, we getThen,For any and , we have . Then, Note thatthen,By Hölder’s inequality, Then,Combining the estimates for , , , and , the proof of Theorem 21 is completed.

4. Proof of Theorem 2

To prove Theorem 2, we need the following analogy of the classical Fefferman-Stein inequality for the sharp maximal function ; its proof can be found in [13].

Lemma 22. Let , , and . If , then we have for all functions such that the left hand side is finite.

Proof. It follows from that ; then, there exists such that . Note that then, by Lemma 22 and Theorem 21, we have Since , by Lemmas 17–20, we get This finishes the proof of Theorem 2.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.