Boundedness for a Class of Generalized Commutators of Fractional Hardy Operators with a Rough Kernel
The authors consider the generalized commutator of fractional Hardy operator with a rough kernel as follows: where , , and with . The authors prove that is bounded on Herz type space and -Central Morrey space with , which is an open problem for .
It is well known that the C-Z singular integrals and their commutators have been studied a lot by many mathematicians; please see  or  for more details. For the generalizations of the commutators of singular integrals, Cohen  studied the following generalized commutator which is defined by where is homogeneous of degree zero and satisfies the moment condition with . Cohen  proved that if and , then is bounded on with . Later, Cohen and Gosselin  considered another type of generalized commutator as follows: where is defined by , the mth remainder of Taylor series of the function at about , and satisfies the following moment condition: with . Obviously, if we choose , becomes , the commutator of generalized by and . Furthermore, becomes if we choose .
Cohen and Gosselin proved that if , and the function has derivatives of order in BMO(), then the operator is bounded on for . Later, was studied by many mathematicians; please see [5, 6] or  for more details. Recently, Wang and Zhang  gave a new proof of Wu’s theorem in  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 . Furthermore, the method used in  is much simpler than that in . Recently, Yu and Tao  proved that is bounded on -Central Morrey space.
Let be a nonnegative integral on : then the Hardy operator is defined by In 1920, Hardy  proved the following inequality: where and the constant is the best possible.
In 2007, Fu et al.  introduced the -dimensional fractional type Hardy operator as follows: where and is a locally integrable function on .
Obviously, when , is just the -dimensional Hardy operator which was proposed by Christ and Grafakos in .
In , the authors gave the characterization of the by the boundedness of the commutator of the fractional type Hardy operator on Herz type spaces. Here the space is defined by the following.
The space can be regarded as the space of bounded mean oscillation, a local version of BMO() at the origin. But the famous John-Nirenberg inequality no longer holds in .
Now we are interested in the following generalized commutator of Hardy operator: where and .
In 2010, Lu and Zhao  proved that when , is bounded on Herz type space and Morrey-Herz type space. Later, Gao and Yu  proved that is bounded on -Central Morrey spaces. However, we would like to point out that the method used in [15, 16] cannot apply to the case when . An interesting question is whether the boundedness of on Herz type space or -Central Morrey space still holds with . In this paper, we will use a different method to answer this question. Furthermore, we will consider the generalized commutator of fractional Hardy operator with a rough kernel as follows: where , , and .
In , we prove that is bounded from to with . Furthermore, we study the endpoint estimates of on spaces with in . In this paper, we will prove that is bounded on Herz type space and -Central Morrey space when .
2. Boundedness of on Herz Type Spaces
In this section, we will give the boundedness of on Herz type spaces. First we introduce some notations that will be used throughout this paper.
Let , , and for : here is the characteristic function of the set .
Definition 2 (see ). Let . Then the homogeneous Herz type space is defined by
where is defined as
with the usual modifications made when or .
Now we show our main results in this section.
Theorem 3. Suppose , with , and has derivatives of order in with . Let with . If , and satisfies the following condition: then there exists a constant , such that
For , is just the commutator of Hardy operator; that is, . We have the following theorem of on Herz type space.
Theorem 4. Suppose with and . Let with . If , , and satisfies (13), then there exists a constant , such that
Lemma 6 (see ). Let , and . If with , then there exists a constant independent of , such that where is defined by By checking  carefully, one can draw the conclusion that if one replaces by , then (16) still holds.
Lemma 7. Let and . If has derivatives of order in with and , then one has where the constant is independent of and .
Proof. From [20, p. 241], we have the following estimates:
where and is the Hardy-Littlewood maximal function of .
Thus we obtain By the above estimates, we can get
For the term , let ; then by the Hölder inequality, Lemma 6, and the boundedness of Hardy-Littlewood maximal function on spaces, we obtain
For the term , let ; then by the Hölder inequality and Lemma 6, we have Combining the estimates of and , we finish the proof of Lemma 7.
Lemma 8 (see ). Let be a function on with th order derivatives in for some . Then where is the cube centered at having diameter .
Lemma 9 (see ). Suppose that , , and ; then
Proof of Theorem 3. To prove Theorem 3, first we split each as
then we have
For the term , we denote ; then it is easy to check . By the fact that , with , we have . As , then by Lemmas 8 and 9, we obtain As and , then by the Hölder inequality, we have As we obtain the following estimates:
For the term , we choose satisfying supp as well as in and we set . Let and . Then it is easy to see that for and with . Thus we get Thus by Lemma 7, we have From [5, p.80], we have the following estimates: where is dependent on . Thus we get As , we obtain the following estimates: For the term , we have When , by condition (13), we get When , by the Hölder inequality and condition (13), we have For the term , we have the following estimates: Combining the estimates of and , we finish the proof of Theorem 3.
3. Boundedness of on -Central Morrey Spaces
In , Wiener gave a way to describe the behavior of a function at the infinity. Later, Beurling  extended Weiner’s idea and introduced a pair of dual Banach spaces, and , with . In , Feichtinger proved that can be described as where is the characterization of the unit ball and is defined as in Section 2.
Now by duality, the space , which is called the Beurling algebra, can be described by
Later, Chen and Lau  as well as García-Cuerva  introduced atomic spaces associated with the Buerling algebra and the dual space of can be described by here the can be regarded as the inhomogeneous central BMO spaces.
In 2000, Alvarez et al.  introduced the -Central bounded mean oscillation space and -Central Morrey space, respectively.
Definition 10 (see ). Given , then a function is said to belong to the -Central bounded mean oscillation space if
Definition 11 (see ). Let and . Then the -Central Morrey space is defined by
From , we know that if , we obtain for and for . Furthermore, when , reduces to and reduces to the space of constant functions. When , coincides with modulo constant and .
In 2011, Fu et al.  proved the boundedness of the commutator of fractional Hardy operator with a rough kernel on -Central Morrey space. Later, Fu et al.  proved the boundness of the weighted Hardy operator and its commutator on -Central Morrey space. In this paper, we will give the boundedness of on -Central Morrey space with .
Our results can be stated as follows.
Theorem 12. Suppose , , with and . Let with , and has derivatives of order in . If , , and , then one has where the constant is independent of and .
For the case , we have the following theorem.
Theorem 13. Suppose , with and . Let with and . If , and , then one has where the constant is independent of and .
Lemma 14 (see ). Suppose , , , and ; then
Proof of Theorem 12. For any , we denote by and by for any . Thus we have the following estimates:
For the term , let . Then it is easy to see . As , then by Lemmas 8 and 9, we have where is the cube centered at and having diameter .
As and with , we have and .
Thus by the Hölder inequality and the condition , we have Note the following fact: