Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 410305, 11 pages
http://dx.doi.org/10.1155/2013/410305
Research Article

Endpoint Estimates for Generalized Commutators of Hardy Operators on Space

1Department of Mathematics, Shangrao Normal University, Shangrao City 334001, China
2Department of Mathematics, Zhejiang University, Hangzhou City 310027, China
3School of Mathematical Science, Beijing Normal University, Beijing City 100875, China

Received 25 March 2013; Accepted 7 May 2013

Copyright © 2013 Xiao Yu and Shanzhen Lu. 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

We study the -boundedness of the generalized commutators of Hardy operator with a homogeneous kernel as follows: , where with , and . We prove that, when , is not bounded from to unless . Finally, we prove that is bounded from to with .

1. Introduction

Let with ; the classical Hardy operator is defined by

A famous result proved by Hardy [1] can be stated as follows;

Hardy [1] also pointed out the fact that the constant in (2) is the best possible. Later, Hardy operator was studied by many mathematicians; please see [2, 3] for more details.

In 1995, Christ and Grafakos [4] studied the following -dimensional Hardy operator; where is the volume of the unit ball in , and they proved the following inequality:

Furthermore, Christ and Grafakos [4] also showed that the constant in (3) is the best possible.

In 2007, Fu et al. [5] considered the following commutator of fractional Hardy operator: where is defined by with . When , we simply denote by and is just the -dimensional Hardy operator proposed by Christ and Grafakos in [4] (without considering the constant ).

In 2011, Fu et al. [6] studied the following -dimensional fractional Hardy operator with a homogeneous kernel: where . Fu et al. [6] proved that is bounded on Herz type space and -central Morrey space. Here is just the commutator of fractional Hardy operator with a homogeneous kernel.

Recently, Zhao et al. [7] gave a counterexample to show that is not bounded from to , and they proved that is bounded from to weak space where denotes the Hardy space.

On the other hand, in 1982, Cohen [8] studied the following generalized commutator: where is homogeneous of degree zero and satisfies the moment condition for . Cohen [8] proved that, if and , then is bounded on with .

Later, Cohen and Gosselin [9] considered another type of generalized commutator as follows: where is defined by , the th remainder of Taylor series of the function at about , and satisfies the following moment conditions: for . Obviously, if we choose , becomes , the commutator of generalized by and .

Cohen and Gosselin [9] proved that is bounded on for if and the function has derivatives of order in . Later, was studied by many mathematicians; for example, see [10, 11] or [12] for more details. Particularly in [11], Lu and Wu studied the endpoint estimates of on space.

It should be pointed out that the generalized commutators of some operators play an important role in the study of partial differential equation. Recently, by using the estimate for the elliptic equation of divergence form with partially BMO coefficients and the boundedness of the Cohen-Gosselin type generalized commutators proved by Yan in [12], Wang and Zhang [13] gave a new proof of Wu’s theorem in [14]. Here we would like to point out that the method used in [13] is much simpler than that in [14].

In this paper, we will consider the following generalized commutator of fractional Hardy operator with a homogeneous kernel: where , and .

As the Hardy operator is controlled by the Hardy-Littlewood maximal function, we have where is defined by

By a simple computation or from [15, pp. 221-222], we have and thus

So we have where

From [15, p. 222], we have the following lemma.

Lemma 1 (see [15]). Let and . If with , and has derivatives of order in BMO, then where the constant is independent of and .

By checking [15, p. 222] carefully, we deduce that (19) still holds if we take . So we have the following proposition.

Proposition 2. Let , with . If and has derivatives of order in BMO, then where the constant is independent of and .

Definition 3 (see [16]). One says a function is an atom if satisfies the following conditions:

It is well known that, if a function belongs to , then it can be written as where each is an atom. Moreover, one has where the infimum is taken over all decompositions of .

Definition 4 (see [17]). A function is said to belong to BMO() if the following sharp maximal function is bounded: where the supreme is taken over all balls and and .

Proposition 5 (see [17]). Let and ; then one has(a);(b).
Obviously, when , can be written as , just the commutator of fractional Hardy operator with a homogeneous kernel.

For the case , and , Lu and Zhao [18] proved that is bounded on Herz type spaces and Morrey-Herz type spaces.

In this paper, we would like to show that is not bounded from to for all . Furthermore, we will prove that is bounded from to , where denotes the weak space. Some ideas of this paper come from Zhao et al. [7].

In this chapter, we would like to show that, if , is not bounded from to .

To show this, let , , and ; then for and , we have and then which indicates that is not bounded from to .

2. Endpoint Estimates for from to

In Section 1, we know that, when , is not bounded from to . In this section, we will prove that, when , is also not bounded from to unless . We have the following conclusions.

Theorem 6. Let , and . Assume that has derivatives of order in ; then the following two statements are equivalent;(i) maps continuously into ;(ii)for any atom supported on certain ball and , there is where with .

In order to prove Theorem 6, we need the following lemma.

Lemma 7 (see [9]). Let be a function on with th order derivatives in for some ; then where is the cube centered at and having diameter .

Proof of Theorem 6. Suppose that is an atom supported on and satisfies (21). Now we take ; then is also an atom and satisfies
Thus by the main results in [19] and the atomic decomposition of the space , it suffices to show that, for any atom , we have .
Let and ; then . For each atom , we split each as
For , taking and so that , it follows from Proposition 2 that
For , as and , we can deduce ; thus we have
Next we denote and ; then by the vanishing condition of , we decompose as follows: where .
For , as , , and , we have
Thus we have
For the term , by Lemma 7, we have
For the term , by the similar estimates of , we obtain
For the term , by the following formula (see [20]): and then together with Lemma 7, we have
Thus for , by the size condition of , we get the following estimates:
For , since , we have the following estimates of :
Thus by the size condition of and Lemma 7, we have
Now we can deduce that is equivalent to . By the vanishing condition of , we can easily get
Consequently, we have finished the proof of Theorem 6.

Next we would like to show that is not bounded from to unless . We have the following theorem.

Theorem 8. Let , and , and assume that has derivatives of order in . Then the following two statements are equivalent:(i) maps continuously into ,(ii) is a polynomial of degree no more than or .

Remark 9. From Theorem 8, we can draw the conclusion that, when , is not bounded from to unless .

Proof of Theorem 8. It is clear that (ii) (i) is obvious. We only need to prove (i) (ii).
Let be an atom supported on the ball , and denote with . By Theorem 6, for any and with , we have
Let , we know . Thus we have
From (44) we can deduce
If , (45) is obviously true. Otherwise, we can easily obtain
Since is arbitrary, must be a constant for each with . So we can deduce that is a polynomial with degree no more than .
Consequently, we have finished the proof of Theorem 8.

3. Boundedness of from to

In Section 2, we prove that, when , is not bounded from to unless . In this section, we will prove that is bounded from to with . Here is defined by

Our results can be stated as follows.

Theorem 10. Suppose that , and . If has derivatives of order in BMO(), then there exists a constant independent of and , such that for any .

For the case , we have the following.

Theorem 11. Let , , and ; then there exists a constant independent of and , such that for any .

Before the proof of Theorems 10 and 11, we need the following lemma.

Lemma 12. Let be defined by where , and ; then is bounded from to .

Proof. For any , we have

Proposition 13. By the proof of Lemma 12 with minor changes, one can draw the conclusion that is bounded from to with .

Proof of Theorem 10. Before giving the proof of Theorem 10, we introduce some notations that are very useful in this section.
For multi-indices , we denote . Furthermore, means that for each , we have . Finally, we denote .
From [19] and by the atomic decomposition of , it suffices to show for any atom , where is defined in Section 2. First we have the following decomposition:
For the term , choosing and with , then by Proposition 2, Hölder inequality, and the size condition of , we have where and .
For the term , as , we have
Then for a fixed , we set . It is easy to check .
Thus by the vanishing condition of and the fact , we can decompose as follows:
Here we can simply denote each by
For , by the fact that and , we have
As for , Lemma 7 in this paper and Lemma  2.2 in [8] tell us that where is a constant only depending on , and is a cube centered at and having diameter .
Thus we obtain
Next we will give the estimates of . First by a cumbersome but straightforward computation, we have
Also noting the fact that where , and is a cube centered at and having diameter .
Thus we obtain
So we have the following estimates of :
Now we get
For , by the vanishing condition of , we can split as follows:
For the term , by Lemma 12 and the size condition of , we obtain
For the term , by the vanishing condition of and the fact , we have
Thus we get