Abstract
Let be the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds, and let be the fractional integrals of for . Assume that is a finite family of locally integrable functions; then the multilinear commutators generated by and are defined by . Assume that belongs to weighted BMO space, ; the authors obtain the boundedness of on weighted Morrey spaces. As a special case, when is the Laplacian operator, the authors also obtain the boundedness of the multilinear fractional commutator on weighted Morrey spaces. The main results in this paper are substantial improvements and extensions of some known results.
1. Introduction and Main Results
Assume that is a linear operator on , which generates an analytic semigroup with a kernel satisfying a Gaussian upper bound; that is,for and all .
The property (1) is satisfied by a large amount of differential operators. One can see [1] for details and examples.
For , the fractional integral generated by the operator is defined by
Let be a finite family of locally integrable functions; then the multilinear commutators generated by and are defined bywhere .
Note that if , which is Laplacian on , then is the classical fractional integral : while is the iterated commutator generated by and : where .
When , it is easy to see that is the commutator generated by and , and when , is the higher commutator.
As we all know, if , the commutator of fractional integral operator is bounded from to , where and (see [2]). In 2004, Duong and Yan [1] generalized the above classical result and obtained the boundedness of the commutator under the same conditions. Simultaneously, the theory on multilinear integral operators and multilinear commutators has attracted much attention as a rapid developing field in harmonic analysis. Mo and Lu [3] studied the boundedness of the multilinear commutators , where , , and .
On the other hand, Muckenhoupt and Wheeden [4] gave some definitions of weighted bounded mean oscillation and obtained some equivalent conditions for them.
Definition 1 (see [4]). Let and be locally integral in and let . A locally integrable function is said to be in if where and the supremum is taken over all balls .
We may note that other weighted definitions for the bounded mean oscillation also have been given by Muckenhoupt and Wheeden in [4].
Definition 2 (see [4]). Let be locally integral in and . A locally integrable function is said to be in if the norm of : satisfies where and the supremum is taken over all balls .
The above two definitions cannot contain each other; throughout this paper, we will make some investigations on the basis of Definition 2.
Recently, Wang [5] obtained some estimates for the commutator on weighted Morrey space (see Definitions 3 and 4), where . Furthermore, Wang and Si [6] obtained the necessary and sufficient conditions for the boundedness of on weighted Morrey spaces when .
Motivated by [1, 3, 5, 6], it is natural to raise the following question: how to establish corresponding boundedness of the multilinear commutator on the weighted Morrey space, where , ?
The question is not motivated only by a mere quest to extend the multilinear commutator from the classical commutator but rather by their natural appearance in analysis (see [3]).
To state the main results, we now give some definitions and notations.
A weight is a locally integrable function on which takes values in almost everywhere. For a weight and a measurable set , we define , the Lebesgue measure of , by and the characteristic function of by . For a real number , ; is the conjugate of ; that is, . The letter denotes a positive constant that may vary at each occurrence but is independent of the essential variable.
Definition 3 (see [7]). Let , ; let be a weight; then weighted Morrey space is defined by where and the supremum is taken over all balls in .
Definition 4 (see [7]). Let , ; let be weight; then two weights weighted Morrey space are defined by where and the supremum is taken over all balls in . If , then we denote for short.
Remark 5. (1) If , , and , then , the classical Morrey space.
(2) If , then , the weighted Lebesgue space; if , , then , the Lebesgue space.
Definition 6 (see [8]). A weight function is in the Muckenhoupt class with if for every ball in , there exists a positive constant which is independent of such that When , , if When , , if there exist positive constants and such that given a ball and is a measurable subset of , then
Definition 7 (see [9]). 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 in .
It is well known that if with , then there exists such that . It follows from Hölder’s inequality that implies for all . Moreover, if , , then we have for some . We thus write to denote the critical index of for the reverse Hölder condition.
Definition 8. The Hardy-Littlewood maximal operator is defined by Let be a weight. The weighted maximal operator is defined by For , , the fractional maximal operator is defined by And the fractional weighted maximal operator is defined by If , we denote for short.
Definition 9. A family of operators is said to be an “approximation to identity” if, for every , is represented by the kernel , which is a measurable function defined on , in the following sense: for every , , for , . Here, is a positive, bounded, decreasing function satisfying for some .
Associated with an “approximation to identity” , Martell [10] introduced the sharp maximal function as follows: where , is the radius of the ball , and for some .
Notice that our analytic semigroup is an “approximation to identity.” In particular, denote
Next, we make some conventions on notation. Given any positive integer , for all , we denote by the family of all finite subsets of of different elements, and, for any , let . Let ; then, for any , we denote , and , and .
In this paper, our main results are stated as follows.
Theorem 10. Assume the condition (1) holds. Let , , , , , and , where denotes the critical index of for the reverse Hölder condition. If , , then
Theorem 11. Let , , , , and , where denotes the critical index of for the reverse Hölder condition. If , , then Moreover, if is the Laplacian, then
Remark 12. We note that our results extend some results in [1, 3]. To be specific, if we take , , and in Theorem 10, it is easy to see that our conclusion is the main result of Duong and Yan [1]. If we only take and in Theorem 10, our result contains the corresponding conclusion of [3].
The remaining part of this paper will be organized as follows. In Section 2, we will give some known results and prove some requisite lemmas. Section 3 is devoted to proving the theorems of this paper.
2. Requisite Lemmas
In this section, we will prove some lemmas and state some known results about weights and weighted Morrey space.
Lemma 13 (see [11]). Let , , and . ThenIn particular, .
Lemma 14 (see [10]). Assume that the semigroup has a kernel which satisfies the upper bound (1). Take , (the set of functions in with bounded support) and a ball such that there exists with . Then, for every , , we can find (independent of ) and constants (which only depend on ) such that where is a fixed constant which depends only on .
As a result, using the above good- inequality together with the standard arguments, we have the following estimates:
For every , , ; if , then In particular, when , , we have
Lemma 15 (Kolmogorov’s inequality; see [9, page 455]). Let , for ; define , , and ; then
Lemma 16 (see [4]). Let . Then the norm of : , is equivalent to the norm of : ; that is, if is a locally integrable function, then is equivalent to
Lemma 17 (see [5]). Let , , , and ; if , , then It also holds for .
Lemma 18 (see [5]). Let , , , and ; if , , , then
Lemma 19 (see [5]). Let , , , , and . Then, for ,
Remark 20. It is easy to see that Lemmas 17, 18, and 19 still hold for .
Lemma 21. Let , , , and ; if , , then
Proof. Since semigroup has a kernel that satisfies the upper bound (1), it is easy to see that, for , . From the boundedness of on weighted Morrey space (see Lemma 17), we get
Remark 22. Since is of weak type, from the above proof, we can obtain that is of weak type.
Lemma 23 (see [1]). Assume the semigroup has a kernel which satisfies the upper bound (1). Then, for , the differential operator has an associated kernel which satisfies
Lemma 24. Assume the semigroup has a kernel which satisfies the upper bound (1), , and . Then, for , , , , andwhere and is the radius of .
Proof. For any , , we haveNoticing that , , from (1), we get Thus,For simplicity, we only consider the case of . We also want to point out that although we state our results on the case of , all results are valid on the multilinear case without any essential difference and difficulty in the proof. So it follows that We split as follows: We now consider the four terms, respectively. Choose satisfing . According to Hölder’s inequality and , we have In the above inequalities, we use the fact that if , then . Thus, the norm of is equivalent to the norm of (see Lemma 16).
For , we first estimate the term that contains . In fact, it follows from the John-Nirenberg lemma that there exist and such that, for any ball and , since . Using the definition of , we getfor some . We can see that (48) yields Thus,On the basis of (50), we now estimate . For the above , select , such that ; by virtue of Hölder’s inequality and , we have Analogous to the estimate of , we also have .
As for , taking advantage of Hölder’s inequality and (50), we get Collecting the estimates of , , , and , it is easy to see that For term , using the fact that (see [12]), we now get By some estimates similar to those used in the estimate for , we conclude that For , using the same method as in dealing with , we get Therefore, Analogously, An argument similar to that used in the estimate for leads to Hence, Next, we will consider the second term in (41). For any , , it is easy to get that and Thus, For simplicity, we also consider the case of . For , similar to the estimate of , we have Noticing that we could use similar methods of the estimates of , , and in the estimates of , , and , respectively. From this together with the fact that (see [12]), it is easy to get Therefore, According to the estimates of and , the lemma has been proved.
Now, we will establish a lemma which plays an important role in the proof of Theorem 10.
Lemma 25. Let , , and ; then, for all , , and , we have
Proof. For any given , take a ball which contains . For , let , . Denote the kernel of by , , where , . Then can be written in the following form:Now expanding asand combining (68) with (69), it is easy to see that where and is the radius of ball .
Take , , and denote ; then We now estimate the above seven terms, respectively. We take as an example; the estimate for the case is the same. For the first term , we split it as follows: Choose such that . Noticing that , by Hölder’s inequality and the similar estimate of , we have For , take that satisfy ; then following Hölder’s inequality and the same idea as that of yieldsTo estimate , applying Lemma 15 (Kolmogorov’s inequality), weak boundedness of (see Remark 22), and Hölder’s inequality, we have We consider the case of , for example, and we split as follows: Take such that ; by virtue of Hölder’s inequality and the same manner as that used in dealing with in Lemma 24, we get For , Therefore, By Lemma 24, we have Next, we consider the term : For , since , , and , it follows that Analogous to the estimate of , we have For , note that , , and . Hence, the estimate of runs as that of yields that For the last term , applying Lemma 23, we have The same manner as that of gives us that According to the above estimates, we have completed the proof of Lemma 25.
3. Proof of Theorems
At first, we give the proof of Theorem 10.
Proof. It follows from Lemmas 13, 14, 25, and 17–21 that Then, we can make use of an induction on to get that This completes the proof of Theorem 10.
Now, we are in the position of proving Theorem 11. If we take in Theorem 10, we will immediately get our desired results.
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 express their gratitude to the anonymous referees for reading the paper carefully and giving some valuable comments. The research was partially supported by the National Nature Science Foundation of China under Grants nos. 11171306 and 11071065 and sponsored by the Scientific Project of Zhejiang Provincial Science Technology Department under Grant no. 2011C33012 and the Scientific Research Fund of Zhejiang Provincial Education Department under Grant no. Z201017584.