It is known that Fourier integral operators arising when solving Schr\"odinger-type operators are bounded on the modulation spaces $\cM^{ p,q}$, for $1\leq p=q\leq\infty$, provided their symbols belong to the Sj\"ostrand class $M^{ \infty,1}$. However, they generally fail to be bounded on $\cM^{ p,q}$ for $p\not=q$. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on $\cM^{ p,q}$ for $p\not=q$, and between $\cM^{ p,q}\to\cM^{ q,p}$, $1\leq q< p\leq\infty$. 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.
Categories: math.FA math.AP
