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# Compactness of Commutators for Singular Integrals on Morrey Spaces

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Published:2011-07-15
Printed: Apr 2012
• Yanping Chen,
Department of Mathematics and Mechanics, Applied Science School, University of Science and Technology Beijing, Beijing 100083, P.R. China
• Yong Ding,
School of Mathematical Sciences, Beijing Normal University,, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education,, Beijing 100875, P.R. China
• Xinxia Wang,
The College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang, 830046, P.R. China
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## Abstract

In this paper we characterize the compactness of the commutator $[b,T]$ for the singular integral operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More precisely, we prove that if $b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO} (\mathbb R^n)$-closure of $C_c^\infty(\mathbb R^n)$, then $[b,T]$ is a compact operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and $0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and $[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$ for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.
 Keywords: singular integral, commutators, compactness, VMO, Morrey space
 MSC Classifications: 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.) 42B99 - None of the above, but in this section

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