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

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.