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Journal of Function Spaces
Volume 2016 (2016), Article ID 1084859, 19 pages
http://dx.doi.org/10.1155/2016/1084859
Research Article

Weighted Estimates for Toeplitz Operators Related to Pseudodifferential Operators

School of Sciences, China University of Mining and Technology, Beijing 100083, China

Received 7 January 2016; Accepted 10 February 2016

Academic Editor: Dashan Fan

Copyright © 2016 Yan Lin 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

The authors establish the weighted estimates for a class of pseudodifferential operators for both cases and , where the weight class is bigger than the classical Muckenhoupt’s weight class. Moreover, the weighted estimates for the Toeplitz operators related to pseudodifferential operators are also obtained. As their special cases, the corresponding results for the commutators of pseudodifferential operators can be deduced.

1. Introduction

The pseudodifferential operator, appearing in 1960s, is formalized in Kohn and Nirenberg [1] and Hörmander [2]. With the development of mathematic theory, the pseudodifferential operator has been a popular mathematic tool, which can be extensively applied in the research of various fields of mathematics, especially in dealing with some basic problems in the PDE.

The classical pseudodifferential operators are defined by Hörmander symbols. Following [3], let , , and be real numbers, and let be a smooth function, defined on such that, for all multi-indices and , the following estimate holds, where is independent of and ; then, is called a symbol in the class . A symbol in is one which satisfies the above estimates for any real number .

It can be deduced that if , if , and if .

Definition 1. The operator given by is called a pseudodifferential operator with symbol , where is a Schwartz function (denote simply by ) and denotes the Fourier transform of .

As usual, will denote the class of pseudodifferential operators with symbols in , and will denote the class of pseudodifferential operators with symbols in .

The boundedness of different kinds of pseudodifferential operators on Lebesgue spaces and weighted Lebesgue spaces has been studied by many authorities. Laptev [4] proved that any pseudodifferential operator in is a standard Calderón-Zygmund operator, so the boundedness on is obvious. Hörmander [3] and Fefferman [5] pointed out that the pseudodifferential operators in are bounded on if and only if , where , , and . Fefferman [5] also obtained the endpoint estimates for the pseudodifferential operators in . Alvarez and Hounie [6] obtained the boundedness of the pseudodifferential operators in on , where , , , and , in which condition is not necessary (also see [7, 8]). Lin and Lu [9] generalized the results in [10] and established the relationship between Lebesgue boundedness and Sobolev boundedness of pseudodifferential operators in related symbol classes.

Miller [11] verified the boundedness of pseudodifferential operators in on weighted spaces whenever the weight function belongs to Muckenhoupt’s class . Tang [12] improved the result of Miller. Following [12], the boundedness of pseudodifferential operators in on weighted spaces was verified whenever the weight function belongs to , where for some , , , and the new class of weight functions is defined as follows.

Definition 2. A nonnegative measurable function is said to be in for , if for every cube in there exists a positive constant independent of such that When , a nonnegative measurable function is said to belong to , if there exists a constant such that, for any cube , Here and in what follows, , and

Denote . Since , for , where denote the classical Muckenhoupt weight class; see [13]. It is acknowledged that if , then is a doubling measure. It follows from the definition and the properties of that if , then may be not a doubling measure. In fact, let ; it is easy to check that is not in and is not a doubling measure, but . This fact implies that

Inspired by Tang [12], a natural question is whether the pseudodifferential operator in is also bounded on weighted spaces with the weight function in .

In , Coifman et al. [14] introduced a class of nonconvolution operators, which is said to be the commutator generated by a singular integral operator and a function :

Auscher and Taylor [15] and Taylor [16] investigated the boundedness of the commutator generated by the pseudodifferential operators in and a suitable function on , Alvarez and Hounie [6] introduced boundedness results for the commutator generated by the pseudodifferential operators in and a function , wherever or , , , , and . Chanillo [7] obtained the boundedness on for the commutator generated by the pseudodifferential operators in and a function , wherever and . Lin [17] partly improved the results in [6, 7] and established the boundedness on for the commutator generated by the pseudodifferential operators in and a function , where , , , , and or , , and Michalowski et al. [18] constructed the boundedness for the commutator generated by the pseudodifferential operators in and a function , where , , , and The boundedness of commutators generated by pseudodifferential operators in and functions on weighted spaces was obtained with the weight functions in in [12].

The boundedness of the Toeplitz operator was introduced by Krantz and Li [19] in . And the commutator of the Calderón-Zygmund operators can be regarded as a special case of the Toeplitz operator , where and are the Calderón-Zygmund operators or ( is the identity operator) and is a multiplication operator. When , Krantz and Li [19] discussed the boundedness of on the homogeneous space.

In this paper, we also concentrated on the weighted Lebesgue boundedness of Toeplitz operators related to pseudodifferential operators and functions. Actually, in this situation, and are pseudodifferential operators in or , and . Moreover, as a special case, we can obtain the corresponding results of the commutators generated by pseudodifferential operators and functions on weighted Lebesgue spaces.

Before stating our main results, we firstly recall some notations and definitions as follows.

Given and , denote the -dilate ball, which is the ball with the same center and radius . Similarly, denote the cube centered at with the side length , where the sides of the cubes are parallel to the coordinate axes. Given a Lebesgue measurable set and a weight , will denote the Lebesgue measure of and . will denote for . We denote by the constants that are independent of the main parameters involved but whose value may differ from line to line. For a measurable set , denote by the characteristic function of . By , we mean that there exists a constant such that .

Definition 3. The Hardy-Littlewood maximal operator is defined by

Definition 4. The maximal operator for is defined by

Definition 5. The dyadic maximal operator for is defined by where denotes any dyadic cube containing in .

A variant of dyadic maximal operator for and is defined by

Definition 6. The dyadic sharp maximal operator for is defined by where denotes any dyadic cube containing in and

A variant of dyadic sharp maximal operator for and is defined by

Definition 7. A function is called a Young function if it is continuous, convex, and increasing and satisfies and as . If is a Young function, we define the -average of a function over a cube by means of the following Luxemburg norm: The generalized Hölder inequality holds, where is the complementary Young function associated with . And we define the corresponding maximal function The example we are going to use in Section 3 is with the maximal function denoted by . The complementary Young function is given by with the maximal function denoted by .
Now we define a function, which will be used in Section 3. It is a negative, Radial, and function of compact support, defined in the -space , with the properties that for and for . Together with , we define another function , by . Then, we have the following partitions of unity of the -space: Let and , respectively, denote the reverse Fourier transform of and ; that is, Write and ; then, and .

The organization of the paper is as follows. The main results are showed in Section 2. Lemmas that we need to prove our main results are accounted for in Section 3. The proof of the main results is in Section 4.

2. Main Results

Now we state our main results. Firstly, we establish the weighted boundedness for the pseudodifferential operators in ,  .

Theorem 8. Suppose , . If , , then is bounded on .

Theorem 9. Suppose , . If , then there exists a constant such that, for any ,

Then, we obtain the weighted estimates for the Toeplitz operators related to pseudodifferential operators and functions.

Theorem 10. Suppose and , or , where , . Let when . If and , , then is bounded on .

Since the commutator can be seen as a special case of the Toeplitz operator, we can get the following result as a corollary of Theorem 10.

Theorem 11. Suppose , . If and , , then is bounded on .

At last, we can also establish the endpoint estimate for the commutator.

Theorem 12. Suppose , . If and , then there exists a constant such that, for any ,

3. Necessary Lemmas

Before giving the proof of our main results, we need some necessary lemmas. In this section, we are inspired by some ideas from [10, 12, 14, 18, 2022].

Lemma 13 (see [12]). Let and . If , then Furthermore, if , then if and only if

Lemma 14. Suppose , and , then there exists a constant such that, for any and ,

Proof. Let be the symbol of ; then, for all real numbers and any multi-indices and , there exists a constant such that Write where First of all, we will prove that, for any , there is a constant , such that, for all , In fact, Now we estimate .
Case 1. If , then by taking in (23), we have Case 2. If , then . Through repeating integration by parts, we can obtain where ; expresses the biggest integer no more than .
Take in (23); then, and Take in (23); then, Thus, .
Next we estimate . Take in (23); then, From the above, we get the inequality we desired. Then, for any , Thus, by taking in (33), we have For any dyadic cube , let . We write Take in (33); then, Taking the supremum of all dyadic cubes containing , we can achieve the inequality Since where denotes any dyadic cube containing in , then which completes the proof of this lemma.

Inspired by the results in [23, 24], we have the following lemma.

Lemma 15. Suppose , , , , and is supported on . If , then there exists a constant such that the inequality holds for all and in and every integer .

Proof.
Case 1. and is an integer.
Let and ; then, Take such that ; then, Case 2. and is not an integer; then, there exists a nonnegative integer such that .
Subcase 1. If , then : Subcase 2. If , then : Thus, we complete the proof of the lemma.

Lemma 16. If , , then, for any and , there exists a constant such that, for any ,

Proof. Let be the symbol of . Firstly, we suppose that has compact -support and prove the various constants that occur in the following arguments will not depend on the support of :For any dyadic cube , let . Decompose .
Case 1. When , take . Since , then Firstly, we estimate . According to [24], we know that is of weak type . By Kolmogorov inequality, we get Secondly, we estimate . Decompose the operator into a sum of simpler operators. Since we can write It follows from that for any ; thus, . By Lemma 14, we have Write Subcase 1. If , then take such that : Now we estimate . Let . Take and in Lemma 15; then, we can get Take and in Lemma 15; then, we have From the above, we know that when ,Subcase 2. If , then also take such that . Write where .
The following facts can be checked easily:(a), where .(b), where .(c) for both and in .(d)If , since , then .Applying Lemma 15, then we have