Riesz potential, Marcinkiewicz integral and their commutators on mixed Morrey spaces

Abstract

This paper deals with the boundedness of integral operators and their commutators in the framework of mixed Morrey spaces. Precisely, we study the mixed boundedness of the commutator $[b,I_\alpha]$, where $I_\alpha$ denotes the fractional integral operator of order $\alpha$ and $b$ belongs to a suitable homogeneous Lipschitz class. Some results related to the higher order commutator $[b,I_\alpha]^k$ are also shown. Furthermore, we examine some boundedness properties of the Marcinkiewicz-type integral $\mu_\Omega$ and its the commutator $[b,\mu_\Omega]$ when $b$ belongs to the $BMO$ class.