Boundedness of Schroedinger type propagators on modulation spaces

TL;DR

This paper investigates conditions under which Schr"odinger-type Fourier integral operators are bounded on modulation spaces for different p,q values, extending known results and utilizing the uncertainty principle.

Contribution

It introduces new conditions on phase and symbol to ensure boundedness of these operators on modulation spaces for p≠q and between different modulation spaces.

Findings

Boundedness on ^{p,q} spaces for pq q under new conditions.

Extension of boundedness results to operators on Wiener amalgam spaces.

Application of the uncertainty principle in the analysis.

Abstract

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.