Time-frequency analysis of fourier integral operators

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.

Boundedness of Fourier Integral Operators on $\mathcal {F}L^p$ spaces

“…If the equation has constant coefficients, then T reduces to a Fourier multiplier and acts continuously on M p without loss of derivatives; cf. [2,4,6]. In the case of variable coefficients, our Theorem 1.2 gives the optimal regularity results on M p .…”

“…version of results in [6]. In Section 4 we prove Theorem 1.2 for a symbol σ(x, η), which, in addition, vanishes for η large.…”

“…This enters the general philosophy of [7]. Figure 1 shows the passage from Hörmander's geometry to Shubin's one, performed in (6). As a motivation of our study we recall that the solutions of the Cauchy problem for a strictly hyperbolic equation can be described by Fourier integral operators.…”

Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

“…The proof of the result in [5], corresponding to Theorem 1.2 with r = d (hence the spatial smooth factorization condition is automatically satisfied) used tools from Time-frequency analysis, relying on our previous work [4]. Instead, the proof of Theorem 1.2 is inspired by more classical arguments in [17].…”

Boundedness of Schrödinger Type Propagators on Modulation Spaces

“…Indeed, FIOs are a mathematical tool to study a variety of problems in partial differential equations, and our FIOs arise naturally in the study of the Cauchy problem for Schrödinger-type operators (see, e.g., [5,7,8,14,17,18]). Basic examples of phase functions within the class under consideration are quadratic forms in the variables x, η (see Example 5.3 below).…”