Sharp estimates of the Schr\"{o}dinger type propagators on modulation spaces

TL;DR

This paper provides sharp estimates for Schrödinger-type propagators on modulation spaces, extending previous results to cases with mild and high phase function growth, using advanced decomposition and scaling techniques.

Contribution

It introduces new boundedness results for Fourier integral operators with symbols in Sjöstrand class on modulation spaces, covering novel growth conditions of phase functions.

Findings

Boundedness on modulation spaces established for operators with symbols in M^{,1}

Results cover both mild and high phase function growth scenarios

Method involves decomposition and scaling ensuring optimality

Abstract

This paper is devoted to conducting a comprehensive and self-contained study of the boundedness on modulation spaces of Fourier integral operators arising when solving Schr\"{o}dinger type operators. The symbols of these operators belong to the Sj\"{o}strand class $M^{\infty,1}$, and their phase functions satisfy certain regularity conditions associated with mixed modulation spaces. Our conclusions cover two novel situations corresponding to the so-called mild and high growth of phase functions. These conclusions represent essential improvements and generalizations of existing results. Our method is based on a reasonable decomposition and scaling of the symbol and phase functions, ensuring their membership in appropriate mixed modulation spaces. In a certain sense, all conclusions of this paper are optimal.