Limiting weak-type behaviors for factional maximal operators and fractional integrals with rough kernel

TL;DR

This paper investigates the limiting weak-type behaviors of fractional maximal operators and fractional integrals with rough kernels, removing smoothness assumptions and extending previous results in harmonic analysis.

Contribution

It establishes limiting weak-type behaviors without smoothness assumptions and characterizes operator boundedness via kernel Lebesgue space membership.

Findings

Extended previous results on fractional operators

Characterized boundedness through kernel Lebesgue space membership

Removed smoothness assumptions on kernels

Abstract

By a reduction method, the limiting weak-type behaviors of factional maximal operators and fractional integrals are established without any smoothness assumption on the kernel, which essentially improve and extend previous results. As a byproduct, we characterize the boundedness of several operators by the membership of their kernel in Lebesgue space on sphere.