Generalized local Morrey spaces and fractional integral operators with rough kernel

Abstract

Let $M_{\Omega,\a}$ and $I_{\Omega,\a}$ be the fractional maximal and integral operators with rough kernels, where $0 < \a < n$. In this paper, we shall study the continuity properties of $M_{\Omega,\a}$ and $I_{\Omega,\a}$ on the generalized local Morrey spaces $LM_{p,\varphi}^{{x_0}}$. The boundedness of their commutators with local Campanato functions is also obtained.