Weighted Sobolev L2 estimates for a class of Fourier integral operators

TL;DR

This paper develops a global calculus for Fourier integral operators with minimal decay assumptions and establishes weighted Sobolev L2 estimates, leading to new smoothing results for hyperbolic PDEs.

Contribution

It introduces a novel global calculus framework and weighted Sobolev estimates for Fourier integral operators, advancing analysis of dispersive PDEs.

Findings

Established global weighted Sobolev L2 estimates for Fourier integral operators.

Derived new smoothing estimates for hyperbolic equations based on data decay.

Provided minimal decay assumptions for phases and amplitudes in the calculus.

Abstract

In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of smoothing estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.