Regularity properties of Schr\"odinger integral operators and general oscillatory integrals

TL;DR

This paper introduces Schr"odinger integral operators and oscillatory integrals, establishing sharp regularity results and applying them to variable-coefficient Schr"odinger equations and other PDEs.

Contribution

It presents new regularity results for Schr"odinger integral operators and general oscillatory integrals, extending classical analysis to inhomogeneous phase functions.

Findings

Sharp local and global regularity results for Schr"odinger integral operators.

Regularity results for oscillatory integrals with inhomogeneous phases.

Optimal local-in-time estimates for variable-coefficient Schr"odinger equations.

Abstract

We introduce the notion of Schr\"odinger integral operators and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of oscillatory integral operators with inhomogeneous phase functions, whose local and global regularity are also established in classical function spaces (both in the Banach and quasi-Banach scales). The results are then applied to obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient Schr\"odinger equations as well as other evolutionary partial differential equations.