On radial and conical Fourier multipliers

TL;DR

This paper explores the relationship between radial and conical Fourier multipliers, providing new endpoint bounds for Bochner-Riesz multipliers on light cones and characterizations of certain convolution inequalities.

Contribution

It establishes a novel connection between radial and conical Fourier multipliers and derives new bounds and characterizations for these operators.

Findings

New weak type endpoint bounds for Bochner-Riesz multipliers

Characterizations of L^p to L^{p,ν} inequalities for radial convolutions

Connections between radial and conical Fourier multipliers

Abstract

We investigate connections between radial Fourier multipliers on $R^d$ and certain conical Fourier multipliers on $R^{d+1}$. As an application we obtain a new weak type endpoint bound for the Bochner-Riesz multipliers associated to the light cone in $R^{d+1}$, where $d\ge 4$, and results on characterizations of $L^p\to L^{p,\nu}$ inequalities for convolutions with radial kernels.