On radial Fourier multipliers and almost everywhere convergence

TL;DR

This paper investigates almost everywhere convergence of Riesz means and related Fourier multipliers on various function spaces, providing new boundedness characterizations and endpoint estimates for these operators.

Contribution

It introduces generalized results for radial Fourier multipliers, characterizes their boundedness on weighted spaces, and establishes sharp endpoint bounds for Stein's square-function.

Findings

Almost everywhere convergence for Riesz means at critical indices.

Boundedness criteria for radial Fourier multipliers on weighted L^2 spaces.

Sharp endpoint bounds for Stein's square-function associated with Riesz means.

Abstract

We study a.e. convergence on $L^p$, and Lorentz spaces $L^{p,q}$, $p>\tfrac{2d}{d-1}$, for variants of Riesz means at the critical index $d(\tfrac 12-\tfrac 1p)-\tfrac12$. We derive more general results for (quasi-)radial Fourier multipliers and associated maximal functions, acting on $L^2$ spaces with power weights, and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformations on weighted $L^2$ spaces, and a sharp endpoint bound for Stein's square-function associated with the Riesz means.