Schrodinger-type propagators, pseudodifferential operators and modulation spaces

We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of non-degenerate generalized quadratic forms that includes Schr¨odinger propagators and pseudodifferential operators. As a byproduct, we obtain a characterization of all exponents p, q, r1, r2, t1, t2 ∈ [1,∞] of modulation spaces such that a symbol in Mp,q(R2d) gives a pseudodifferential operator that is continuous from Mr1,r2 (Rd) into Mt1,t2 (Rd).