A Characterization of Central BMO Space via the Commutator of Fractional Hardy Operator

This paper is devoted in characterizing the central BMO ðRnÞ 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 ðRnÞ 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 ðRnÞ space by exploiting the center symmetry of the Hardy operator deeply.


Introduction
In this paper, we focus on the need for characterizing the central BMO ðℝ n Þ space via the boundedness of the commutators of the following fractional Hardy operators and S n−1 = fx ∈ ℝ n : |x| = 1g denotes the unit sphere in ℝ n . For a function b, the commutators of H Ω,α and H * Ω,α can be written as In [1], Fu et al. considered the boundedness of H Ω,α and ½b, H Ω,α on homogeneous Herz spaces and Lebesgue spaces under the assumption that Ω satisfies (2) and (4). We recall the results from ( [1], Proposition 3.1 and Theorem 3.1) as Suppose that Then, H Ω,α is bounded from L p 1 ℝ n ð Þto L p 2 ℝ n ð Þ; b ; H Ω,α ½ is bounded from L p 1 ℝ n ð Þto L p 2 ℝ n ð Þ: ( For the boundedness of the classical fractional Hardy operator, see [2]. For a ball B r ≔ Bð0, rÞ ⊂ ℝ n (i.e., a ball centered at 0 with radius r > 0) and p ≥ 1, CBMO p ðℝ n Þ is the central BMO ðℝ n Þ function space, which was introduced by Lu and Yang [3] It is easy to see that CBMO ðℝ n Þ can be understood as a local version of the classical BMO ðℝ n Þ space at the origin [4] and Hence, the famous John-Nirenberg inequality for BMO ðℝ n Þ space is not true for CBMO ðℝ n Þ 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][6][7][8][9][10][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 Ω. Under the premise that Ω is smooth enough, i.e., Ω ∈ C ∞ ðS n−1 Þ or Ω ∈ Lip 1 ðS n−1 Þ, see, for example, [12][13][14][15]. It is difficult to weaken the smoothness of Ω, Chen and Ding [16] considered a characterization of BMO ðℝ n Þ space under the condition that Ω satisfies the following Hölder condition of log type It is easy to check that (10) is weaker than the Lipschitz condition and stronger than the condition (4). For q ≥ 1 and Ω ∈ L q ðS n−1 Þ, we call Ω satisfies the L q -Dini condition if Ð 1 0 ððw q ðδÞÞ/ðδÞÞ < ∞, where w q ðδÞ is defined as As a useful supplement of [1], we give a characterization of the CBMO ðℝ n Þ space via the boundedness of ½b, H Ω,α and ½b, H * Ω,α as follows.
A part of Theorem 1 has been proven in [1], we will show the rest of Theorem 1 in Section 2. In what follows, we will denote C by a positive constant which may vary from line to line. The symbol A ≲ B means A ≤ CB and ℤ for the set of all integers. Last, but not least, B r ≔ Bð0, rÞ,

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 Ω, which plays a key role in the proof.
2 Journal of Function Spaces
(b) If Ω furthermore satisfies the L q≥1 -Dini condition, then, there is a C > 0 such that for 0 < C < 1/2, r > 0, x ∈ ℝ n with jxj < Cr; we have Proof. We give the proof by a slight modification from [17].
For jxj ≥ 4jyj, we first show that Indeed, the first inequality can be obtained immediately from ( [18], 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 L q -Dini condition as and the estimate follows from ( [19], 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 ðL p 1 ðℝ n Þ, L p 2 ðℝ n ÞÞ boundedness for H Ω,α is contained in ([1], Proposition 3.1), it is enough to check the boundedness for H * Ω,α . Namely, the task is now left to show that there exist constants C > 0 such that for any f ∈ L p 1 ðℝ n Þ, one has To do so, we first recall a useful estimates from [1] as 3 Journal of Function Spaces Applying Hölder's inequality to q, p 1 , s for s > 1 and (24), we have which is our desired result. Now, we can prove Theorem 1. Without loss of generality, we can assume that kbk CBMO max fp 2 ,sg ðℝ n Þ = 1 in the proof of (a) since b ∈ CBMO max fp 2 ,sg ðℝ n Þ. We see at once that the ðL p 1 ðℝ n Þ, L p 2 ðℝ n ÞÞ boundedness of ½b, H Ω,α is just ( [1], 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 J 2 , we see at once that Journal of Function Spaces The Hölder inequality, along with (24), implies The term J 22 need a further decomposition as follows: Applying the Hölder inequality, we deduce From the fact that the term J 222 can be estimated as follows: which is the desired result and (a) is obtained. Next, we verify (b) inspired by [18]. Namely, we need to show that there is a constant CðΩ, α, p, nÞ such that For abbreviation, we assume that k½b, H Ω,α k L p 1 →L p 2 = 1, k½b, H * Ω,α k L p 1 →L p 2 = 1, and b B r = 0 since ½b − b B r , H Ω,α = ½b, H Ω,α . Let f ðyÞ = sgn ðbðyÞÞχ B r ðyÞ: It is easy to check that ð1/jB r jÞ Ð ℝ n f ðyÞbðyÞdy = Θ: Applying (3) and (10), we deduce that for A and γ appearing in (10), there is a constant C 1 < 1 such that σðDÞ > 0 for D = fx′ ∈ S n−1 : Ωðx′Þ ≥ ðð2AÞ/ððlog ð2/C 1 ÞÞ γ ÞÞg: For x ′ ∈ D and y ′ ∈ S n−1 with jx ′ − y ′ j ≤ C 1 , we obtain from (10) that Writing E = fx ∈ ℝ n : jxj > C 2 r and & x′ ∈ Dg with C 2 = 3C −1 1 + 1 > 4, we see at once that for x ∈ E, j½b, H Ω,α f ðxÞj ≥ jH Ω,α ðbf ÞðxÞj − jbðxÞjjH Ω,α f ðxÞj ≔ K 1 ðxÞ − K 2 ðxÞ: We conclude from (16) that jðx − yÞ′ − x′j ≤ 3jyj/jxj ≤ C 1 since jyj < r and jxj > C 2 jyj > 4jyj, and hence, Ωððx − yÞ′Þ ≥ A/ ðlog ð2/C 1 ÞÞ γ , and finally, that 5 Journal of Function Spaces Furthermore, For abbreviation, we write This, along with the estimates for K 1 and K 2 , one has Consequently, This in turn implies that C 6 Θ p 2 r n ≤ |F| + ðC 2 rÞ n : Thus, According to F ⊂ E, C 2 > 4, jyj > C 2 jxj, Lemma 2 and (37), we can obtain that for x ∈ B r and y ∈ F, We continue to choose f * ðyÞ = ðsgnðbðyÞÞÞχ F ðyÞ for x ∈ B r and get It is easy to check that To deal with the term L 1 ðxÞ, we first obtain from (44) that Then, the estimate for L 1 ðxÞ consists of two cases.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The author declares that there are no conflicts of interest.