The metaplectic action on modulation spaces

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