Radial Fourier Multipliers in $\mathbb{R}^3$ and $\mathbb{R}^4$

TL;DR

This paper establishes new $L^p$ bounds for radial Fourier multipliers in three and four dimensions, using geometric intersection estimates to characterize boundedness and restricted strong type properties.

Contribution

It provides the first $L^p$ characterization of radial Fourier multipliers in four dimensions and extends bounds for three-dimensional multipliers using geometric intersection techniques.

Findings

Radial Fourier multipliers in $ ext{R}^3$ are restricted strong type (p,p) for $1<p<13/12$.

Radial Fourier multipliers in $ ext{R}^4$ are bounded on $L^p$ if and only if their Fourier transform is in $L^p$, for $1<p<36/29$.

Geometric bounds on intersections of annuli are used to control tangencies and prove multiplier bounds.

Abstract

We prove that for radial Fourier multipliers $m: \mathbb{R}^3\to\mathbb{C}$ supported compactly away from the origin, $T_m$ is restricted strong type (p,p) if $K=\hat{m}$ is in $L^p(\mathbb{R}^3)$, in the range $1<p<\frac{13}{12}$. We also prove an $L^p$ characterization for radial Fourier multipliers in four dimensions; namely, for radial Fourier multipliers $m: \mathbb{R}^4\to\mathbb{C}$ supported compactly away from the origin, $T_m$ is bounded on $L^p(\mathbb{R}^4)$ if and only if $K=\hat{m}$ is in $L^p(\mathbb{R}^4)$, in the range $1<p<\frac{36}{29}$. Our method of proof relies on a geometric argument that exploits bounds on sizes of multiple intersections of three-dimensional annuli to control numbers of tangencies between pairs of annuli in three and four dimensions.