Mixed radial-angular integrabilities for commutators of fractional Hardy operators with rough kernels

Abstract

This paper is devoted to studying the boundedness of commutators \(\textrm{H}_{\Omega ,\beta }^b\) generated by the rough fractional Hardy operators \(\textrm{H}_{\Omega ,\beta }\) with the symbol b on the mixed radial-angular spaces. When b is a mixed radial-angular central bounded mean oscillation function and \(\Omega \in L^s(S^{n-1})\) for some \(s>1\), the boundedness of \(\textrm{H}_{\Omega ,\beta }^b\) on the mixed radial-angular homogeneous Herz spaces is established. Meanwhile, the boundedness for \(\textrm{H}_{\Omega ,\beta }^b\) on the mixed radial-angular homogeneous \(\lambda \)-central Morrey spaces is also obtained, provided that b belongs to the mixed radial-angular homogeneous \(\lambda \)-central bounded mean oscillation spaces and \(\Omega \in L^s(S^{n-1})\) for some \(s>1\).