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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 568202, 13 pages
Weighted Estimates for Commutators of -Dimensional Rough Hardy Operators
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China
Received 3 December 2012; Revised 21 January 2013; Accepted 5 February 2013
Academic Editor: Lars Diening
Copyright © 2013 Zhuanxi Ren and Shuangping Tao. 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.
We establish the weighted estimates for the commutators and which are generated by the -dimensional rough Hardy operators and central BMO functions on the weighted Lebesgue spaces, the weighted Herz spaces and the weighted Morrey-Herz spaces. Furthermore, the weighted Lipschitz estimates are also obtained.
The classical Hardy operator and its adjoint operator are defined, respectively, by Hardy proved the following most celebrated inequality in : Moreover, where . Hardy’s inequality has received considerable attention. In 1995, Christ and Grafakos obtained it on . Firstly, we recall that the definitions of the -dimensional Hardy operator and its adjoint operator given by Christ and Grafakos are as follows: Let , . Then Christ and Grafakos’s results in  are where and the constant is the best.
Let be a nonnegative integrable function on . The -dimensional rough Hardy operator and its adjoint operator are defined, respectively, by where , , is homogeneous of degree zero. Then the commutators generated by or and a locally integrable function are defined, respectively, as follows:
In , it was proved that the commutators were bounded on the Lebesgue spaces and Herz spaces if . Recently, Gao obtained in  that is also bounded from the Morrey-Herz spaces to if . On the other hand, Gao and Wang in  had established the weighted estimates on weighted Lebesgue and Herz-type spaces for the commutators and which are generated by -dimensional Hardy operator and and . It is easy to see that and when . A natural question is whether commutators of -dimensional standard rough Hardy operators and also have boundedness on these weighted spaces. The answer is affirmative. The main purpose of this paper is to generalize the above results on the weighted Lebesgue spaces, the weighted Herz spaces, and the weighted Morrey-Herz spaces.
First let us recall some standard definitions and notations before introducing our main results. The classical weighted theory was first introduced by Muckenhoupt in the study of weighted boundedness of the Hardy-Littlewood maximal functions in . A weight is a locally integrable function on which takes values in at almost everywhere. Let denote the ball with the center and radius . is a constant which may vary from line to line. For , let and . We use to denote the characteristic functions of the set . We also denote the weighted measure of by ; that is, . And let be the conjugate index of whenever , .
Definition 1. We say that , , if where is a positive constant which is independent of the choice of .
For the case , , if A weight function if it satisfies the condition for some .
Definition 2 (see ). Let , , , and and a weighted function. Then the homogeneous weighted Herz space is defined by where with the usual modification made when or .
Obviously, when , ; when , , .
Definition 3 (see ). Let , , , , and and a weighted function. Then the homogeneous weighted Morrey-Herz space is defined by where with the usual modification made when or .
Obviously, when , ; when , .
Definition 4 (see ). Let , , and a weighted function. We say that a locally integrable function belongs to the weighted Lipschitz space if where .
The smallest bound satisfying conditions above is then taken to be the norm of in this space and is denoted by . We also put . Obviously, for the case , the space is the classical Lipschitz space . If , then García-Cuerva in  proved that the space coincide for any and the norms of are equivalent with respect to different values of . That is .
Definition 5 (see ). Let , and a weighted function. A function is said to belong to the weighted central BMO space if where .
Obviously, . When , .
2. Main Results
Now, we present our main results as follows.
Theorem 6. Let , , , and with homogeneous of degree zero for some . Suppose that and are defined by (9) and ; then and are bounded from to .
Theorem 7. Let , , , , and with homogeneous of degree zero for some . Suppose that ; then(a) is bounded from to if ;(b) is bounded from to if .
Theorem 10. Let , , , and with homogeneous of degree zero for some . Suppose that and , then(a) is bounded from to if ;(b) is bounded from to if .
Theorem 11. Let , , , , and with homogeneous of degree zero for some . Suppose that for ; then and are bounded from to .
Theorem 12. Let , , , , , and with homogeneous of degree zero for some . Suppose that for ; then(a) is bounded from to if ;(b) is bounded from to if .
Theorem 14. Let , , , , , and with homogeneous of degree zero for some . Suppose that for and ; then(a) is bounded from to if ;(b) is bounded from to if .
Remark 15. As , our results are consistent with the main results in .
3. Proofs of the Main Results
Lemma 16 (see ). Let . Then there exist constants , , and depending only on -constant of , such that for any measurable subset of a ball ,
Remark 17. If , it is easy to see from Lemma 16 that there exists a constant and such that as and as .
Lemma 18 (see ). Let and . Then there exists a constant such that, for any ,
Lemma 19 (see ). Let and . Then there exists a constant such that, for any ,
Lemma 20 (see ). Let . Then, for any , where .
We are now in a position to give the proof of Theorem 7.
Proof of Theorem 7. (a) From the definition in (9), we readily see that
For , noticing that and then by Lemma 20 and Hölder’s inequality it follows that When , by inequality (23) and Remark 17, we have When , , also by inequality (23) and Remark 17 we can obtain that Hence, for , applying inequalities (24) and (25), we have Now we turn our attention to the estimate of . According to the conditions, we can further decompose as Below we shall give the estimates of . By Hölder’s inequality, we can deduce When , using Lemma 20, we can obtain the following result by inequality (28) and Remark 17 that When , , noticing inequality (28) and then by using Hölder’s inequality, Lemma 20, and Remark 17, we can obtain that Hence, for , by estimates (29) and (30), we get that Now, let us deal with the last term . When , noting inequality (23), by using Lemma 18, Lemma 20, and Remark 17, we have that When , , by using Lemma 18, inequalities (21) and (23), Remark 17, and an application of Hölder’s inequality give us that As before, for , using inequalities (32) and (33), we can get Summing up the above estimates for , , and , it is true that
Consequently, We shall consider two cases. As , noticing that , we can deduce For the case of , it follows from Hölder’s inequality and that Therefore, we conclude the proof of Theorem 7(a).
(b) Following the same procedure as that of Theorem 7(a), we can also show the conclusion of (b); the details are omitted here.
Proof of Theorem 10. We only give the proof of (a) as . The proof of (b) is similar to that for (a). By using (15) and the above estimates for , , and as those of the proof of Theorem 7, we can obtain that This completes the proof of Theorem 10.
Proof of Theorem 12. (a) Let . We follow the strategy of the proof of Theorem 7; we can also write For , when , noticing that and , by inequality (23) and Remark 17, we obtain that As , , observing that and , by the inequality (23) and Remark 17, again using Hölder’s inequality, we can obtain that Therefore, for , by estimates (41) and (42), we obtain Now we come to estimate the other term . Again, we shall further decompose as We first turn to deal with the term . By Hölder’s inequality, we can obtain that As , noticing that and , we can deduce from inequality (45), Lemma 20, and Remark 17 that As , observing that and and then by using Hölder’s inequality, inequality (45), Lemma 20, and Remark 17, we can obtain that Hence, for , by combining inequalities (46) and (47), we have Finally, let us deal with the term . When , noticing that , by using Lemma 19, inequality (23), and Remark 17, we can get that When , observing that , it follows from Hölder’s inequality, Lemma 19, inequality (23), and Remark 17 that