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

Yan Lin, Zongguang Liu, Chengdan Xu, and Zichu Ren

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, 20–22].

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

Putting the two subcases together, we get

Case 2. When , write ; then,

Similar to the estimates for , by Kolmogorov inequality, we get

By Lemmas 14 and 15, we have

From the above we know that when the symbol of has compact -support, the inequality

holds.
When the symbol of has no compact -support, let ; then, as and . Suppose is the symbol of the pseudodifferential operator ; then, . Through the dominated convergence theorem, we can achieve for all . And by the application of the dominated convergence theorem, we get, for each cube ,

as . Applying the previous results to and taking the limit as , it is easy to get that

if , and

if . Thus, the proof of this lemma is completed.

Lemma 17 (see [12]). If , , , then there exists a constant such that

Lemma 18 (see [23]). Let , , , and

(a)Let be doubling; that is, for ; then there exists a constant depending upon the condition of and doubling condition of such that

for every function such that the left-hand side is finite.(b)Let , ; then,

Lemma 19. Let ; then, there exists a constant such that

Proof. For any dyadic cube , we consider two cases about .
Case 1. Consider . Since , then

Case 2. Consider . We have

Thus, we can obtain

Lemma 20. Let and ; then, there exists a constant such that

Proof. For any dyadic cube , it follows from that From Lemma 19, we have

Lemma 21 (see [12]). Let and . For any and , there exist constants and such that

for all , where

If the pseudodifferential operator is given with symbol , we can also write it as ; that is,

where .

Write , where , ; then,

where and

Lemma 22. Let , ; then, for all multi-indices and , the following estimates

hold for every integer , where is independent of

Proof. Write

When , for supported in and , by taking , we have

When , for supported in and , by taking , we have

From all the above, we can obtain that, for any ,

Lemma 23. Let , , and be the kernel of ; then, for all , the following inequalities hold:

Proof. Write

Put ; then,

It follows from Lemma 22 that, for every integer ,

Case 1. When ,

First, we estimate . Take and is the biggest integer no more than ; then,

Next, we estimate . By taking , we have

From the above we can obtain that when ,

Case 2. When , by taking , we get

Thus, for all ,

Similarly we can obtain that, for all ,

Thus, we complete the proof of the lemma.

Lemma 24 (see [23]). If and , then

Lemma 25. Let ; then, there exists a constant such that

Proof. For all , by Lebesgue differential theorem and Lemma 24, we have

Then, taking supremum for all , we can have the estimates

Lemma 26 (see [12]). Let and ; then, there exist constants and , which are independent of and , such that

Lemma 27. Let , , , or , , , , and . Suppose when . Then, for all ,

Proof. For fixed , let any dyadic cube and . Since , , then

Write

Case 1. When , take :

Firstly, we estimate :

Subcase 1. When , , according to [4, 24], we know that is of the weak type . By Kolmogorov inequality, we get

Subcase 2. When , take ; let ; then, and

Since and , then . We have

By the John-Nirenberg inequality of , we can get that

Thus,

Secondly, we estimate :

Let
Subcase 1. When , , we can express that

where and is the symbol of . According to Lemma 23, we know that satisfies the following inequality:

for all .
Take such that . We get

Now we estimate . Let and . If is on the line between and , then and . Thus,

Then, we can get

Applying the generalized Hölder inequality,

and noticing that for , we have

Since , then by Lemma 25 we have

Thus,

Next we estimate . Take ; similar to the estimate for , we get

Then, we achieve that

By the generalized Hölder inequality, we get

Similar to the estimates for , we can get

From the above, we can obtain

So

Subcase 2. When , we have

Thus,

So when ,

Case 2. When , write ; then,

Subcase 1. When , , since is of the weak type , by Kolmogorov inequality and the generalized Hölder inequality, we can get

Write

Let , Then, by taking in Lemma 23, we have

By the generalized Hölder inequality, we can get

By Lemma 25, we have

Thus,

Then,

Subcase 2. When , take ; then, and :

Since , then . Write

Thus,

So when ,

Concluding the two cases, we can get

Lemma 28. Let , , , , and . Then, for any ,

The proof is similar to that of Lemma 27; thus, we omit the details here.

Lemma 29 (see [12]). Let , , and . Then, there exists a constant such that, for any function and ,

Lemma 30. Let , , , , , and . Then, there exists a constant such that, for any ,

Proof. Taking , by Lebesgue differential theorem, we have a.e. for

Thus, we get

Then, we only need to show that

Let

By Lemma 21, for any and , there is

Let ; by Lemma 28, we have

Thus, we can obtain

Let ; then, . Let ; then, . Since, for any and , , then and . Dividing by in both sides of the above inequality, we get

Then, we can achieve that

Then,

By Lemmas 16, 18(a), and 25, denoting , we have

It follows from Lemma 21 that . Take such that ; then,

From the above, the following inequality holds:

4. Proof of Main Results

Now we are able to prove our main results.

Proof of Theorem 8. Take ; then, for all ,

Taking , by Lemmas 16–18, we get

Proof of Theorem 9. For any and , we have

Thus,

Taking in Lemma 18 and applying Lemmas 13 and 16, we can get

Thus, for any , the desired inequality holds.

Proof of Theorem 10. Take and . Then, , and

It follows from Lemmas 16–18, 20, and 26-27 and Theorem 8 that

Thus, is bounded on .

Proof of Theorem 12. Let . By homogeneity, we need only to show that

Taking , by Lemma 29-30, we get

Thus, the desired result holds.

Competing Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11171345), Beijing Higher Education Young Elite Teacher Project (YETP0946), the Fundamental Research Funds for the Central Universities (2009QS16), and the State Scholarship Fund of China.

References

J. J. Kohn and L. Nirenberg, “An algebra of pseudo-differential operators,” Communications on Pure and Applied Mathematics, vol. 18, pp. 269–305, 1965.

L. Hörmander, “Pseudo-differential operators,” Communications on Pure and Applied Mathematics, vol. 18, pp. 501–517, 1965.

L. Hörmander, “Pseudo-differential operators and hypoelliptic equations,” Proceeding of Symposia in Pure Mathematics, vol. 10, pp. 138–183, 1967.

A. A. Laptev, “Spectral asymptotics of a class of Fourier integral operators,” Trudy Moskovskogo Mathematics Obshchestva, vol. 43, pp. 92–115, 1981 (Russian).

C. Fefferman, “Lp-bounds for a class of pseudo-differential operators,” Israel Journal of Mathematics, vol. 14, pp. 413–417, 1973.

J. Alvarez and J. Hounie, “Estimates for the kernel and continuity properties of pseudo-differential operators,” Arkiv för Matematik, vol. 28, no. 1, pp. 1–22, 1990.

S. Chanillo, “Remarks on commutators of pseudo-differential operators,” Contemporary Mathematics, vol. 205, pp. 33–37, 1997.

S. Chanillo and A. Torchinsky, “Sharp function and weighted L p estimates for a class of pseudo-differential operators,” Arkiv för Matematik, vol. 24, pp. 1–25, 1986.

Y. Lin and S. Z. Lu, “Pseudo-differential operators on Sobolev and Lipschitz spaces,” Acta Mathematica Sinica, vol. 26, no. 1, pp. 131–142, 2010.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series no. 30, Princeton University Press, Princeton, NJ, USA, 1970.

N. Miller, “Weighted sobolev spaces and pseudo-differential operators with smooth symbols,” Transactions of the American Mathematical Society, vol. 269, no. 1, pp. 91–109, 1982.

L. Tang, “Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators,” Journal of Functional Analysis, vol. 262, no. 4, pp. 1603–1629, 2012.

J. Garcia-Cuerva and J. R. de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, The Netherlands, 1985.

R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Journal of Functional Analysis, vol. 103, pp. 611–635, 1976.

P. Auscher and M. E. Taylor, “Paradifferential operators and commutator estimates,” Communications in Partial Differential Equations, vol. 20, no. 9-10, pp. 1743–1775, 1995.

M. Taylor, “Commutator estimates,” Proceedings of the American Mathematical Society, vol. 131, no. 5, pp. 1501–1507, 2003.

Y. Lin, “Commutators of pseudo-differential operators,” Science in China Series A: Mathematics, vol. 51, no. 3, pp. 453–460, 2008.

N. Michalowski, D. J. Rule, and W. Staubach, “Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4183–4209, 2010.

S. G. Krantz and S. Y. Li, “Boundedness and compactness of integral operators on spaces of homogeneous type and applications,” Journal of Mathematical Analysis and Applications, vol. 258, no. 2, pp. 629–641, 2001.

R. Coifman, “A real variable characterization of ,” Studia Mathematica, vol. 51, pp. 269–274, 1974.

C. Fefferman and E. M. Stein, “ spaces of several variables,” Acta Mathematica, vol. 129, no. 3-4, pp. 137–193, 1972.

M. Paluszyński, “Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss,” Indiana University Mathematics Journal, vol. 44, no. 1, pp. 1–17, 1995.

C. Pérez, “Endpoint estimates for commutators of singular integral operators,” Journal of Functional Analysis, vol. 128, no. 1, pp. 163–185, 1995.

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.