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International Journal of Analysis
Volume 2013 (2013), Article ID 798528, 12 pages
http://dx.doi.org/10.1155/2013/798528
Research Article

New Weighted Norm Inequalities for Pseudodifferential Operators and Their Commutators

1Department of Mathematics, Ho Chi Minh City University of Pedagogy, Ho Chi Minh City 70000, Vietnam
2Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia

Received 18 August 2012; Accepted 13 November 2012

Academic Editor: Serge Nicaise

Copyright © 2013 The Anh Bui. 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

This paper is dedicated to study weighted inequalities for pseudodifferential operators with amplitudes and their commutators by using the new class of weights and the new BMO function space BMO which are larger than the Muckenhoupt class of weights and classical BMO space BMO, respectively. The obtained results therefore improve substantially some well-known results.

1. Introduction and the Main Results

For a pseudodifferential operator given formally by where the amplitude satisfies certain growth conditions. The boundedness of pseudodifferential operators has been studied extensively by many mathematicians; see, for example, [17] and the references therein. One of the most interesting problems is studying the weighted norm inequalities for pseudodifferential operators and their commutators with BMO function; see, for example, [59].

In this paper we consider the following classes of symbols and amplitudes (in what follows we set ).

Definition 1. Let and , and .(a)We say when for each triple of multi-indices , , and there exists a constant such that (b)We say when for each triple of multi-indices , , and there exists a constant such that

Definition 2. Let and , and .(a)We say when for each pair of multi-indices and there exists a constant such that (b)We say when for each multi-indices there exists a constant such that

It is easy to see that , , , and . The classes and were studied in [3, 8]. For further information about these two classes, we refer the reader to, for example, [3, 10]. The class was introduced by [11], and it is the natural generalization of the class . This class is much rougher than that considered in [6, 7]. The amplitude class in Definition 1 is rough in the variable, but smooth in the variable. This is smaller than the class introduced in [5] but includes the class .

The aim of this paper is to study the weighted norm inequalities for pseudodifferential operators and their commutators by using the new BMO functions and the new class of weights. Firstly, we would like to give brief definitions on the new class of weights and the new BMO function space (we refer to Section 2 for details).

The new classes of weights for , where , , is the set of those weights satisfying for all ball . We denote that . It is easy to see that the new class is strictly larger than the Muckenhoupt class . Indeed, for example, the weight with belongs to the class , but it is not in , for , see, for example, [12].

The new BMO space with is defined as a set of all locally integrable functions satisfying where and . A norm for , denoted by , is given by the infimum of the constants satisfying (12). Clearly for and . We define .

Our main result is the following theorem.

Theorem 3. Let with or . If is bounded on for all , then(a) is bounded on for and ;(b)for any , the commutator bounded on for and .
In particular, the obtained results in (a) and (b) still hold for with .

We would like to specify some applications of Theorem 3.

In [8], the author studied the weighted inequalities of when the symbol belongs to the class with . It was proved that is bounded on for , . Recently, the author in [9] showed that and its commutator with a BMO function are bounded on for and by the different approach. Here, by using Theorem 3, we not only reobtain the boundedness of on for and but also obtain the new result on the boundedness of its commutator with BMO functions.

Corollary 4. Let . Then we have the following:(i) is bounded on for and ;(ii)for each , the commutator is bounded on for and .
In particular, the obtained results in (i) and (ii) still hold for with .

Now we consider the class . If with and , then the authors in [5] proved that the pseudodifferential operator and its commutators with BMO functions are bounded on for and ; see [5, Theorems 3.3 and 4.5]. So, Theorem 3 leads us to the following result.

Corollary 5. Let with and . Then we have the following:(i) is bounded on for and ;(ii)for each , the commutator is bounded on for and .
In particular, the obtained results in (i) and (ii) still hold for with .

It was proved in [5, Theorem 3.7] that if with and , then and are bounded on for and with . Therefore, in the light of Theorem 3, we have the following:

Corollary 6. Let with and . Then we have the following:(i) is bounded on for and ;(ii)for each , the commutator is bounded on for and .
In particular, the obtained results in (i) and (ii) still hold for with .

For smooth amplitudes, we have the following result.

Corollary 7. Let with . Then we have the following:(i) is bounded on for and ;(ii)for each , the commutator is bounded on for and .
In particular, the obtained results in (i) and (ii) still hold for with .

Proof. The remark in [1, page 11] tells us that is bounded on for . Thanks to Theorem 3, we conclude that and are bounded on for and .

The outline of the paper is as follows. In Section 2, we first recall some definitions of the new class of weights and the new BMO function spaces . Then we also review some basic properties concerning and . Section 3 represents some kernel estimates for the pseudodifferential operator . The proof of the main result will be given in Section 4.

2. Preliminaries

To simplify notation, we will often just use for and for the measure of for any measurable subset . Also given , we will write for the -dilated ball, which is the ball with the same center as and with radius . For each ball we set that

2.1. The New Class of Weights and New Function Spaces

Recently, in [12], a new class of weights associated to Schrödinger operators , where the potential , the reverse Hölder class has been introduced. According to [12], the authors defined the new classes of weights for , where , , is the set of those weights satisfying for all ball . We denote that , where the critical radius function is defined by

In this paper, we consider the particular case when . In this situation the new classes of weights are defined by for , where is the set of those weights satisfying for all ball . We denote that .

It is easy to see that the new class is larger than the Muckenhoupt class . The following properties hold for the new classes ; see [12, Proposition 5].

Proposition 8. The following statements hold:(i) for ,(ii)if with , then there exists such that . Consequently, .

Similarly, by adapting the ideas to [13], the new space with is defined as a set of all locally integrable functions satisfying where and . A norm for , denoted by , is given by the infimum of the constants satisfying (12). Clearly for and . We define .

The following result can be considered to be a variant of John-Nirenberg inequality for the spaces .

Proposition 9. Let . If , then for all balls
(i)
(ii) for all .

The proof is similar (even easier) to [13, Lemma 1 and Proposition 3] and hence we omit details.

2.2. Weighted Estimates for Some Localized Operators

A ball of the form is called a critical ball if . We have the following result.

Proposition 10. There exists a sequence of points in so that the family of critical balls where , satisfies the following:(i),(ii)there exists a constant such that for any , .

Note that the more general version of Proposition 10 is obtained by [14]. However, in our particular situation, for convenience, we would like to give a simple proof of this proposition.

Proof. Let us consider the family of balls . Using Vitali covering lemma, we can pick the subfamily of balls so that is pairwise disjoint and where . This gives (i).
To prove (ii), pick any . Let be the set of all indices so that . Note that if , then . Therefore, for all . Since is pairwise disjoint, . This is equivalent to that . Hence, . This completes our proof.

We consider the following maximal functions for and : where .

Also, given a ball , we define the following maximal functions for and : where .

We have the following lemma.

Lemma 11. For , let be a sequence of balls as in Proposition 10. Then for all and .

Proof. We adapt the argument in [13, Lemma 2] to our present situation.
By Proposition 10, we have It can be verified that, for , . Note that since is supported in , operators and are Hardy-Littlewood and sharp maximal functions defined in viewed as a space of homogeneous type with the Euclidean metric and the Lebesgues measure restricted to . Moreover, by definition of , if , then , where , and is the class of Muckenhoupt weights on the spaces of homogeneous type . Moreover, due to [12, Lemma 5], for all . Therefore, using Proposition 3.4 in [15] gives To complete the proof, we need only to check that for . We have If , due to , . Hence, in this situation, we have Otherwise, if , it is obvious that . So we have This completes our proof.

Let . For and , we define the following functions for and : where .

When , we write instead of . The following result gives the weighted estimates for .

Proposition 12. Let and , . Then we have provided that .

Without loss of generality, we assume that . Assume that . For , . This implies that where .

Let be the family of critical balls given by Proposition 10. Note that if , where . These estimates and Hölder’s inequalities give Since , by definition of the classes , we have

This together with (26) gives This completes our proof.

For a family of balls given by Proposition 10, we define the operator , as where and with being the Hardy-Littlewood maximal function. We have the following result.

Proposition 13. If and , , then is bounded on .

Proof. We have For each , if we consider as a space of homogeneous type with the Euclidean metric and the Lebesgues measure restricted to , then . Moreover, it can be verified that and the constant is independent of .
Therefore, by (ii) of Proposition 10, This completes our proof.

3. Some Kernel Estimates

Let be a smooth radial function which is equal to on the unit ball centered at origin and supported on its concentric double. Set and . Then, we have and supp for all . Moreover, for any multi-index and , we have

Lemma 14. Let with , and . Let for .(a)For each , (b)If with and , then, for each , there exist so that for any ball , , and , so that (c)If , , then there exist so that for any ball , , and , so that as long as ; and as long as .

Proof. We refer to Lemma 3.1 in [5] for the proof of (a).
(b) We first note that since , we have
Since , and , we have . If , using (a) with so that gives This together with the fact that gives where .
If , we have
We will claim that, for all , we have
Indeed, we have for all integers , Using integration by parts, we get that We write If , . Therefore, in this situation, Otherwise, . This together with (39) gives Therefore, The general statement for noninteger values of follows by interpolation of the inequality for and , where . Therefore, (43) holds for all . Now taking so that , we have
It remains to take care of the term . Repeating the previous arguments we also obtain At this stage, using the mean value theorem (applied for each component of ) and then using the definition of the class give for all integer . Hence, by interpolation again, for all . Repeating the arguments used to estimate , we conclude that
Therefore, LHS . It remains to show that To do this, we repeat the arguments above with . Since the proof of this part is analogous to (55), and hence we omit details here. This completes our proof.
(c) If , using the argument as in (b), we have
If , we have The previous arguments in (b) show that Hence, By taking and repeating the previous arguments, we obtain that This completes the proof of (c).

Since the associated kernel of the operator is given by with as in Lemma 14, from Lemma 14 we deduce directly the following result.

Lemma 15. Let with or , , and let be the associated kernel of the operator , the conjugate of .(a)For any , we have (b)For any , there exists so that any ball , , , , we have

4. Proof of Theorem 3

Note that, by duality argument, the linear operator is bounded on , if and only if its conjugate is bounded on . Moreover, by Hölder’s inequality, it can be verified that if and only if . Therefore, it suffices to prove (a) and (b) for and with . Before coming to the proof of Theorem 3, we need the following results.

Lemma 16. Let with or , , and , . If is bounded on for all , then for any and there exists such that for all balls ,
(a)
(b)

Proof. (a) We split where . For each , we have Using Hölder’s inequality and the fact that is bounded on , , we write For the term we have, for , Applying (a) of Lemma 15, we have This completes the proof of (a).
(b) Taking , we write So, we have We now take care of . By Hölder’s inequality, we can write where with .
Due to boundedness of , one has
To estimate , using (69) gives .
The estimate for can be proceeded in the same method. Indeed, we write where and .
To estimate , using Hölder’s inequality, we have
For the term , due to (a) of Lemma 15, we can write
By Hölder’s inequality and Proposition 9, we give
From (77) and (76) we obtain that
This completes our proof.

Remark 17. The result in Lemma 16 still holds if we replace the critical ball by .

Lemma 18. Let with or , and , . If is bounded on for all , then for any and there exists so that, for all and with , we have
(a)
(b)

Proof. (a) Using (b) of Lemma 15, we write where is the smallest integer so that .
To estimate , let and be families of balls as in (29). If , then for all