Boundedness of Schroedinger type propagators on modulation spaces

It is known that Fourier integral operators arising when solving Schr"odinger-type operators are bounded on the modulation spaces ${\text{cM}}^{p,q}$, for $1\le p=q\le \mathrm{\infty }$, provided their symbols belong to the Sj"ostrand class $M^{\mathrm{\infty },1}$. However, they generally fail to be bounded on ${\text{cM}}^{p,q}$ for $p≠q$. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on ${\text{cM}}^{p,q}$ for $p≠q$, and between ${\text{cM}}^{p,q}\to {\text{cM}}^{q,p}$, $1\leq q< p\leq\infty$. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.