Generalized local Morrey spaces and fractional integral operators with rough kernel

TL;DR

This paper investigates the boundedness and continuity of fractional maximal and integral operators with rough kernels on generalized local Morrey spaces, including their commutators with local Campanato functions, expanding understanding of their functional analysis properties.

Contribution

It introduces new boundedness results for fractional operators with rough kernels on generalized local Morrey spaces and analyzes their commutators with local Campanato functions.

Findings

Boundedness of $M_{\, ext{Omega}, ext{ extalpha}}$ and $I_{ ext{ extOmega}, ext{ extalpha}}$ on $LM_{p, ext{ extphi}}^{x_0}$.

Continuity properties of these operators established.

Boundedness of their commutators with local Campanato functions demonstrated.

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.