On a theorem of M. Jodeit Jr. on pushforwards of Fourier multipliers

TL;DR

This paper generalizes a classical theorem about Fourier multiplier symbols and their pushforwards from Euclidean spaces to the broader context of locally compact groups, including non-abelian ones.

Contribution

It characterizes the continuous homomorphisms of locally compact groups that preserve the Fourier multiplier property under pushforward, extending previous abelian results.

Findings

Generalization of Jodeit's theorem to non-abelian groups

Characterization of homomorphisms preserving Fourier multiplier symbols

Analysis of pushforwards of positive definite distributions

Abstract

A classical theorem of M. Jodeit Jr. implies that if a compactly supported distribution on $\mathbf{R}^d$ is the symbol of an $L^p(\mathbf{R}^d)$-$L^q(\mathbf{R}^d)$ Fourier multiplier, then its pushforward by the canonical homomorphism from $\mathbf{R}^d$ to $\mathbf{T}^d$ is the symbol of an $\ell^p(\mathbf{Z}^d)$-$\ell^q(\mathbf{Z}^d)$ Fourier multiplier. In the present work, we generalise this result to the setting of locally compact groups, including those non-abelian, by characterising the continuous homomorphisms of locally compact groups by which, for every $p,q\in[1,\infty]$, the pushforward of a compactly supported distribution symbol of an $L^p$-$L^q$ Fourier multiplier is a symbol of the same type as those which are open. Motivated by a simple proof in the abelian case, we also investigate pushforwards of positive definite distributions.