Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020View this Special Issue
Research Article | Open Access
Lei Zhang, Shaoguang Shi, "A Characterization of Central BMO Space via the Commutator of Fractional Hardy Operator", Journal of Function Spaces, vol. 2020, Article ID 3543165, 7 pages, 2020. https://doi.org/10.1155/2020/3543165
A Characterization of Central BMO Space via the Commutator of Fractional Hardy Operator
This paper is devoted in characterizing the central BMO space via the commutator of the fractional Hardy operator with rough kernel. Precisely, by a more explicit decomposition of the operator and the kernel function, we will show that if the symbol function belongs to the central BMO space, then the commutator are bounded on Lebesgue space. Conversely, the boundedness of the commutator implies that the symbol function belongs to the central BMO space by exploiting the center symmetry of the Hardy operator deeply.
In this paper, we focus on the need for characterizing the central BMO space via the boundedness of the commutators of the following fractional Hardy operators
is the dual operator of . Here, satisfies and denotes the unit sphere in .
For a function , the commutators of and can be written as
In , Fu et al. considered the boundedness of and on homogeneous Herz spaces and Lebesgue spaces under the assumption that satisfies (2) and (4). We recall the results from (, Proposition 3.1 and Theorem 3.1) as
For the boundedness of the classical fractional Hardy operator, see . For a ball (i.e., a ball centered at with radius ) and , is the central BMO function space, which was introduced by Lu and Yang  via the norm
It is easy to see that CBMO can be understood as a local version of the classical BMO space at the origin  and
Hence, the famous John-Nirenberg inequality for BMO space is not true for CBMO space, which reveals that they have quite different properties.
In the study of harmonic analysis, the characterization of function spaces via the boundedness of the commutators plays an important role in the field of PDEs, see, for example, [5–11] and the references therein. However, there are less attention paid for the commutators with rough kernels since the characterization depends heavily on the smoothness of the kernel function . Under the premise that is smooth enough, i.e., or , see, for example, [12–15]. It is difficult to weaken the smoothness of , Chen and Ding  considered a characterization of space under the condition that satisfies the following Hölder condition of log type
As a useful supplement of , we give a characterization of the CBMO space via the boundedness of and as follows.
Theorem 1. Let . (a)If then (b)if
A part of Theorem 1 has been proven in , we will show the rest of Theorem 1 in Section 2. In what follows, we will denote by a positive constant which may vary from line to line. The symbol means and for the set of all integers. Last, but not least, , , , and with .
2. Proof of Theorem 1.1
We prove Theorem 1 in this section. To do so, we need one lemma about the estimates of the kernel function , which plays a key role in the proof.
Proof. We give the proof by a slight modification from . For , we first show that
Indeed, the first inequality can be obtained immediately from (, Lemma 2). Since
(a) can be shown by (2) and (10) as
We are left to show (b). The first estimate can be obtained directly by the –Dini condition as
and the estimate
follows from (, p.65-77).
Accordingly, the second can be deduced similarly. In fact, which is the desired one.
The following is the boundedness of the fractional Hardy operators.
Proof. Since the boundedness for is contained in (, Proposition 3.1), it is enough to check the boundedness for . Namely, the task is now left to show that there exist constants such that for any , one has To do so, we first recall a useful estimates from  as for , and . Since Applying Hölder’s inequality to for and (24), we have which is our desired result.
Now, we can prove Theorem 1. Without loss of generality, we can assume that in the proof of (a) since . We see at once that the boundedness of is just (, Theorem 3.1). To complete the proof of (a), what is left is to show is that
It is easy to check that
Using Hölder’s inequality, we have
For the term , we see at once that
The Hölder inequality, along with (24), implies
The term need a further decomposition as follows:
Applying the Hölder inequality, we deduce
From the fact that the term can be estimated as follows: which is the desired result and (a) is obtained.
Next, we verify (b) inspired by . Namely, we need to show that there is a constant such that
For abbreviation, we assume that , , and since . Let It is easy to check that
Writing we see at once that for , We conclude from (16) that since and , and hence, , and finally, that
For abbreviation, we write
This, along with the estimates for and , one has
This in turn implies that Thus, is proved if . If , we see immediately that
We continue to choose for and get
It is easy to check that
To deal with the term , we first obtain from (44) that
Then, the estimate for consists of two cases.
Case 1. . Since for , we conclude from (47) and that
Case 2. . In this case, This in turn reveals that