Description
To any sufficiently regular distribution $m$ on a locally compact group is associated, by the mean of the Fourier transform, some sort of « differential operator with symbol $m$ » on the dual of this group which in general is merely a quantum group. In 1970, M. Jodeit Jr. has shown that if a compactly supported distribution on $\mathbf R^d$ is the symbol of a continuous linear operator from $L^p(\mathbf R^d)$ to $L^q(\mathbf R^d)$, then its pushforward by the canonical homomorphism from $\mathbf R^d$ to $\mathbf T^d$ is the symbol of a continuous linear operator from $\ell^p(\mathbf Z^d)$ to $\ell^q(\mathbf Z^d)$. We propose a generalisation of this result by characterising the continuous homomorphisms of locally compact groups by which, for all exponents $p$ and $q$, the pushforward of a compactly supported distribution symbol of a continuous linear operator from $L^p$ to $L^q$ is again the symbol of a continuous linear operator from $L^p$ to $L^q$ as being those which are open.
