No abstract

We use time-frequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames [6,27], the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudodifferential operators with symbols in M ∞,1 [25], for some unimodular Fourier multipliers [2] and metaplectic operators [10,23].2000 Mathematics Subject Classification. 35S30,47G30,42C15.

We construct a one-parameter family of algebras F IO(Ξ, s), s ≥ 0, consisting of Fourier integral operators. We derive boundedness results, composition rules, and the spectral invariance of the operators in F IO(Ξ, s). The operator algebra is defined by the decay properties of an associated Gabor matrix around the graph of the canonical transformation.2000 Mathematics Subject Classification. 35S30, 47G30, 42C15.

We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V (x) = ±|x| 2 .

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L 2 of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [