Compactness of Commutators for Singular Integrals on Morrey Spaces

Chen, Yanping ; Ding, Yong ; Wang, Xinxia

Canadian journal of mathematics, Tome 64 (2012) no. 2, pp. 257-281

Résumé

In this paper we characterize the compactness of the commutator $\left[ b,\,T \right]$ for the singular integral operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ . More precisely, we prove that if $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$ , the $\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ -closure of $C_{c}^{\infty }\left( {{\mathbb{R}}^{n}} \right)$ , then $\left[ b,\,T \right]$ is a compact operator on the Morrey spaces ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for $1\,<\,p\,<\,\infty $ and $0\,<\,\lambda \,<\,n$ . Conversely, if $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\left[ b,\,T \right]$ is a compact operator on the ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ for some $p\,\left( 1\,<\,p\,<\,\infty\right)$ , then $b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$ . Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $\left[ b,\,T \right]$ on ${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$ 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.

Détail

DOI : 10.4153/CJM-2011-043-1

Mots-clés : 42B20, 42B99, singular integral, commutators, compactness, VMO, Morrey space