Compactness of Commutators of Riesz Potential on Morrey Spaces

Abstract

In this paper, the authors give a characterization of the (L p, λ , L q, λ )-compactness for the Riesz potential commutator [b , I α ]. More precisely, the authors prove that the commutator [b , I α ] is a compact operator from the Morrey space

Characterization of Functions via Commutators of Bilinear Fractional Integrals on Morrey Spaces

“…Therefore, our results essentially extend the corresponding ones in [5,6]. In addition, Theorem 1.3 can also be regarded as the generalization of the corresponding result in [9] from the linear setting to the multilinear setting.…”

“…Employing the ideas of [6,9], our approach is as follows: Under the assumption of that [b, Assume that b ∈ BM O(R n )with b * = 1. Then there exist ǫ > 0 and a sequence of cubes {Q j (y j , d j )} ∞ j such that for every j,…”

“…The boundedness and compactness of [b, I α ] i on variant function spaces have been the topic in many articles recently, see [1,2,3,5,6,12,13,17,18,19,26], among numerous references. One of the interesting questions on [b, I α ] i is whether it can be used to characterize BM O(R n ) by boundedness, or CM O(R n ) by compactness, as those in the linear setting (see [7,9,27] etc). Recently, Chaffee [5] established the following result.…”

“…Also, the boundedness for the classical operators in harmonic analysis and the compactness of the commutators for such classical operators in L p,λ (R n ) were extensively studied, for examples see [8,9,10,11,14,17,28] and references therein. In particular, Ding and Mei [17] established the following boundedness and compactness of [b, I α ] i for i = 1, 2 in Morrey spaces.…”

“…In Section 2, we will prove Theorem 1.1 and the proof of Theorem 1.2 will be given in Section 3. We remark that our ideas are greatly motivated by [5,6,9,16].…”