Weighted BMO Estimates for Toeplitz Operators on Weighted Lebesgue Spaces

Abstract

The authors establish the weighted BMO estimates for a class of Toeplitz operators related to strongly singular Calderón-Zygmund operators on weighted Lebesgue spaces. Moreover, the corresponding result for the Toeplitz operators related to classical Calderón-Zygmund operators can be deduced.

1. Introduction

The introduction of the strongly singular Calderón-Zygmund operator is motivated by a class of multiplier operators with symbols given by away from the origin, where and . Fefferman and Stein [1] enlarged the multiplier operators onto a class of convolution operators. Coifman [2] also considered a related class of operators for . The strongly singular nonconvolution operator was introduced by Alvarez and Milman [3], whose properties are similar to those of the classical Calderón-Zygmund operator, but the kernel is more singular near the diagonal than that of the standard case.

Definition 1. Let be a bounded linear operator. is called a strongly singular Calderón-Zygmund operator if the following conditions are satisfied.(1) can be extended into a continuous operator from into itself.(2)There exists a function continuous away from the diagonal such that  if for some and . , for with disjoint support.(3)For some , both and its conjugate operator can be extended into continuous operators from to , where .

Alvarez and Milman [3, 4] discussed the properties of the strongly singular Calderón-Zygmund operator on Lebesgue spaces. Lin [5] proved the boundedness of the strongly singular Calderón-Zygmund operator and its commutators on Morrey type spaces. The authors in [6] established the weighted estimates for the commutator of the strongly singular Calderón-Zygmund operator on weighted Morrey spaces.

The commutator generated by the Calderón-Zygmund operator and a locally integrable function can be regarded as a special case of the Toeplitz operator , where and are the Calderón-Zygmund operators or ( is the identity operator), . When , Krantz and Li [7] discussed the boundedness of on the homogeneous spaces. The commutator generated by the fractional integral operator and a locally integrable function can be regarded as a special case of another class of Toeplitz operators , where are the Calderón-Zygmund operators or , and are the bounded linear operators on , , and is the fractional integral operator. When , Qiu [8] obtained the boundedness of on the homogeneous spaces from to , where . Moreover, Lin and Lu [9] discussed the boundedness of the two kinds of Toeplitz operators and related to strongly singular Calderón-Zygmund operators and functions on Lebesgue spaces.

In this paper, we are interested in the Toeplitz operators related to strongly singular Calderón-Zygmund operators and weighted functions. Actually, in this situation, are the strongly singular Calderón-Zygmund operators or , and are the bounded linear operators on weighted Lebesgue spaces, , and is a weighted function. And we will focus on the boundedness of this kind of Toeplitz operators on weighted Lebesgue spaces.

Before stating our main results, first we need to recall some definitions and notations as follows.

Definition 2. A nonnegative measurable function is said to be in the Muckenhoupt class with if, for every cube in , there exists a positive constant independent of such that where denotes a cube in with the side parallel to the coordinate axes and . When , a nonnegative measurable function is said to belong to , if there exists a constant such that, for any cube , Denote . It is well known that if with , then for all and for some .

Definition 3. A weighted function belongs to the reverse Hölder class if there exist two constants and such that the following reverse Hölder inequality holds for every cube in .

It is well known that if with , then there exists a such that . It follows directly from Hölder’s inequality that implies for all .

Definition 4. Suppose that is a nonnegative locally integrable function on . Define , if there exists a constant , such that, for any cube in ,

Definition 5. Let and let be a weighted function. A locally integrable function is said to be in the weighted space if where and the supremum is taken over all cubes . Moreover, we denote simply by when .

Definition 6. The Hardy-Littlewood maximal operator is defined by We set , where .
The sharp maximal operator is defined by where .
Let be a weight. The weighted maximal operator is defined by We also set , where . The fractional maximal operator and the weighted case are defined by respectively.

2. Main Results

Now we state our main results as follows.

Theorem 7. Let be strongly singular Calderón-Zygmund operators or , let be the same as in Definition 1, and . Suppose , , , , and   with , , ; and are bounded operators on . When , . Then is bounded from to .

If we consider the extreme cases and in Definition 1, then the strongly singular Calderón-Zygmund operator comes back to the classical Calderón-Zygmund operator. Thus, we can get the boundedness of the Toeplitz operators related to classical Calderón-Zygmund operators on weighted Lebesgue spaces as a corollary of Theorem 7.

Corollary 8. Let be classical Calderón-Zygmund operators or . Suppose , , , , and with , , ; and are bounded operators on . When , . Then is bounded from to .

3. Lemmas

Before giving the proof of our main results, we need some necessary lemmas.

Lemma 9. If , then , where .

Lemma 10. If , then , where .

The results of Lemmas 9 and 10 can be deduced directly by Definitions 2 and 4 and Hölder’s inequality. We omit the details.

Lemma 11. If , then and , where .

Proof. Since , then by Lemma 10. It follows from , , and Lemma 9 that . Then applying Lemma 10, we have .

Lemma 12 (see [6]). Let and let be a function in . Suppose , , and . Then

Lemma 13 (see [6]). Given , there is , for all .

Lemma 14 (see [10]). The fractional integral operator is bounded from to , where , , and .

Lemma 15 (see [11, 12]). Let . Then, for any , there exists an absolute constant such that .

Lemma 16 (see [3, 4]). If is a strongly singular Calderón-Zygmund operator, then can be defined to be a continuous operator from to and is of weak type.

By Lemma 16, Definition 1, and the interpolation theory, we can get that is bounded on , . Besides the -boundedness, the strongly singular Calderón-Zygmund operator still has another kind of boundedness properties on Lebesgue spaces. If we interpolate between and , where is given as in Definition 1 and , then is bounded from to with and . It is easy to see that in this situation. Then we interpolate between and weak to obtain the boundedness of from to , where and . In this situation, if and only if . In a word, the boundedness properties of the strongly singular Calderón-Zygmund operator on Lebesgue spaces can be summarized as follows.

Remark 17. The strongly singular Calderón-Zygmund operator is bounded on for . And is bounded from to , , and . In particular, if we restrict in of Definition 1, then is bounded from to , where and .

Lemma 18 (see [13]). Let and . Then .

Lemma 19. If and , then, for any , we have

Proof. By Lemma 18, we have

Lemma 20. If , , and , then .

Proof. For any ball which contains , we have Since then Thus we can get the desired conclusion by taking over all balls which contain .

Lemma 21 (see [13]). Suppose that , , , and . Then .

Lemma 22 (see [14]). Suppose that , , , and . Then

Lemma 23 (see [14]). Suppose that , , and . If , then where is independent of .

Lemma 24 (see [14]). Let , . If is such that , then

Lemma 25. Let be strongly singular Calderón-Zygmund operators or , let be the same as in Definition 1, and . Suppose , , and with , , and . Suppose also , when . Then for a.e. we have

Proof. Denote , where For , let be a ball centered at with radius . Then We consider first. Write Then Since and , then . There is such that . Denote . By Lemma 14 we can get that is bounded from to . Then Since , there is an such that . Let . Since , we have ,  ,  , and . By Hölder’s inequality, we get Note that ; a direct calculation shows that Applying Hölder’s inequality to , we can get Thus Secondly, we consider in the following two cases.

Case 1 (). Since and , then and there is satisfying . By Remark 17, we can find such that the strongly singular Calderón-Zygmund operator is bounded from to and . So there is such that . Let be a ball centered at whose radius is . Consider Then Write If is a strongly singular Calderón-Zygmund operator, then Since , there is such that . Let ; then . Applying Lemma 12, we have The fact implies that . According to Lemma 13, we get If , then By (36) and (37), we have Write If is a strongly singular Calderón-Zygmund operator, then The fact implies that . According to Lemma 13, we get Note that ; if , then Thus, So

Case 2 (). Consider Then Since , there is satisfying . Then the strongly singular Calderón-Zygmund operator is bounded from to . And there is such that . Let . We have , , , and . Then Write If is a strongly singular Calderón-Zygmund operator, then Since when , then Thus, So In conclusion, combining the two cases, we can get the desired estimate for .

4. Proof of Main Result

Now we are able to prove our main result.

Proof of Theorem 7. Since , we have . So ; that is, . There exists such that . Since , then . Applying Lemmas 24 and 25, we have Since and , then . It follows from and that and . Then and by Lemma 11. Applying Lemmas 19, 20, and 22, we have Since , by Lemma 11. Applying Lemma 21, we have It follows from and Lemma 23 that Therefore, This completes the proof of Theorem 7.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for helpful suggestions and comments. This work was supported by the National Natural Science Foundation of China (11171345), Beijing Higher Education Young Elite Teacher Project (YETP0946), and the Fundamental Research Funds for the Central Universities (2009QS16).