On the Commutators of Singular Integral Operators with Rough Convolution Kernels

Guo, Xiaoli ; Hu, Guoen

Canadian journal of mathematics, Tome 68 (2016) no. 4, pp. 816-840

Résumé

Let ${{T}_{\Omega }}$ be the singular integral operator with kernel $\left( \Omega \left( x \right) \right)/{{\left| x \right|}^{n}}$ , where $\Omega $ is homogeneous of degree zero, has mean value zero, and belongs to ${{L}^{q}}\left( {{S}^{n-1}} \right)$ for some $q\,\in \,\left( 1,\,\infty\right]$ . In this paper, the authors establish the compactness on weighted ${{L}^{p}}$ spaces and the Morrey spaces, for the commutator generated by $\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ function and ${{T}_{\Omega }}$ . The associated maximal operator and the discrete maximal operator are also considered.

Détail

DOI : 10.4153/CJM-2015-044-1

Mots-clés : 42B20, 47B07, commutator, singular integral operator, compact operator, completely continuous operator, maximal operator, Morrey space