Schr\"odinger type propagators, pseudodifferential operators and modulation spaces

TL;DR

This paper establishes continuity properties of Fourier integral operators with symbols in modulation spaces, including Schr"odinger propagators and pseudodifferential operators, and characterizes the exponents for boundedness between modulation spaces.

Contribution

It provides new continuity results for Fourier integral operators with symbols in modulation spaces and characterizes the exponents for their boundedness between modulation spaces.

Findings

Continuity results for Fourier integral operators with modulation space symbols.

Characterization of exponents for bounded pseudodifferential operators.

Includes Schr"odinger propagators within the class of phase functions.

Abstract

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 nondegenerate generalized quadratic forms that includes Schr\"odinger propagators and pseudodifferential operators. As a byproduct we obtain a characterization of all exponents $p,q,r_1,r_2,t_1,t_2 \in [1,\infty]$ of modulation spaces such that a symbol in $M^{p,q}(\mathbb R^{2d})$ gives a pseudodifferential operator that is continuous from $M^{r_1,r_2}(\mathbb R^d)$ into $M^{t_1,t_2}(\mathbb R^d)$.