Endpoint bounds of square functions associated with Hankel multipliers

TL;DR

This paper establishes endpoint bounds for square functions linked to Hankel multipliers on radial functions, leading to sharp multiplier theorems and bounds for associated maximal operators.

Contribution

It provides the first endpoint bounds for these square functions and introduces a sharp Marcinkiewicz-type theorem for multivariate Hankel multipliers.

Findings

Endpoint bounds for square functions with Hankel multipliers

Sharp Marcinkiewicz-type multiplier theorem established

L^p bounds for maximal operators generated by Hankel multipliers

Abstract

We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^{p}$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^p$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrig\'{o}s and Seeger for characterizations of Hankel multipliers.