The weak type $(1,p)$ for convolution operators on locally compact groups

TL;DR

This paper establishes necessary and sufficient conditions for the weak (1,p) boundedness of convolution operators on locally compact groups, extending classical results and applying to Fourier multipliers on Lie groups.

Contribution

It generalizes classical results on convolution operators to the setting of locally compact groups and provides new criteria for weak (1,p) boundedness.

Findings

Characterization of weak (1,p) boundedness conditions

Extension of classical results to locally compact groups

Applications to Fourier multipliers on Lie groups

Abstract

In this paper we provide necessary and sufficient conditions for the $\textnormal{weak}(1,p)$ boundedness, $1< p<\infty,$ of convolution operators on locally compact (Hausdorff) topological groups. So, we generalize a classical result due to Sobolev-Hardy-Littlewood and Stepanov. Applications to Fourier multipliers on Lie groups also are given.