Characterization for commutators of n-dimensional fractional Hardy operators

Abstract

In this paper, it was proved that the commutator H β,b generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L p 1 (R n ) to L p 2 (R n ) if and only if b is a CṀO(R n ) function, where 1/p1 − 1/p2 = β/n, 1 < p1 < ∞, 0 β < n. Furthermore, the characterization of H β,b on the homogenous Herz spaceK α,p q (R n ) was obtained.

Commutators of weighted Hardy operators on $\mathbb {R}^{n}$

“…The problem of whether the -dimensional Hardy operator can characterize the central mean oscillation space has been proved by Fu, Liu, Lu and Wang [7]. The result in [7] can be regarded as a generalization of Long and Wang's result [12] in the higher-dimensional case. Another is the weighted Hardy operator which is the object of study in this paper.…”

A note on commutators of Hardy operators

“…In [7], [8], Fu and Lu et al obtained the following result. Lemma 2.2 Let β ≥ 0, 0 < p 1 ≤ p 2 < ∞, 1/q 1 − 1/q 2 = β/n, 1 < q 1 < ∞, 1/q 1 + 1/q 1 = 1, b ∈ CṀO max(q2 ,q 1 ) (R n ).…”

“…Recall that for a locally integrable function f in R n , the n-dimensional fractional Hardy operator H β [7] is defined as follows. For 0 ≤ β < n H β f (x) = 1 |x| n −β |y |< |x| f (y) dy, x ∈ R n \{0}.…”

“…For example, there is no John-Nirenberg lemma for CṀO. Motivated by [11] and [14], the authors in [7] gave a new characterization of the CṀO space via the boundedness of H β ,b on the Herz space.…”

“…The question is whether the condition b ∈ CṀO s in [8] together with [7] is optimal. The purpose of this note is to give a counterexample to show that the index s with respect to CṀO is sharp.…”