The metaplectic action on modulation spaces

We study the mapping properties of metaplectic operators $\widehat{S}\in \mathrm{M}\mathrm{p}(2d,\mathbb{R})$ on modulation spaces of the type ${\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)$. Our main result is a full characterisation of the pairs $(\widehat{S},{\mathrm{M}}^{p,q}({\mathbb{R}}^d))$ for which the operator $\widehat{S}:{\mathrm{M}}^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}^{p,q}({\mathbb{R}}^d)$ is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that $\widehat{S}$ is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of $\widehat{S}:{\mathrm{M}}^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}^{p,q}({\mathbb{R}}^d)$ transfers to $\widehat{S}:{\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)$.