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

Abstract

We prove that for radial Fourier multipliers $m:{ℝ}^3\to ℂ$ supported compactly away from the origin, $T_m$ is restricted strong type $\left(p,p\right)$ if $K=\widehat{m}$ is in $L^p\left({ℝ}^3\right)$, 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 $m:{ℝ}^4\to ℂ$ supported compactly away from the origin, $T_m$ is bounded on $L^p\left({ℝ}^4\right)$ if and only if $K=\widehat{m}$ is in $L^p\left({ℝ}^4\right)$, 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 3-dimensional annuli to control numbers of tangencies between pairs of annuli in three and four dimensions.