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Compactness of Commutators for Singular Integrals on Morrey Spaces


Published:20110715
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 precompact set,
which has interest in its own right.