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

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 ${\mathrm{\ell }}^p({\mathbf{Z}}^d)$-${\mathrm{\ell }}^q({\mathbf{Z}}^d)$ 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 $p,q\in [1,\mathrm{\infty }]$, 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.