Local smoothing and Hardy spaces for Fourier integral operators on manifolds

TL;DR

This paper develops Hardy spaces tailored for Fourier integral operators on manifolds, leading to sharp local smoothing estimates that enhance understanding of wave equations and nonlinear problems on curved spaces.

Contribution

It introduces Hardy spaces for Fourier integral operators on manifolds and derives sharp local smoothing estimates applicable to wave equations and nonlinear problems.

Findings

Sharp local smoothing estimates for Fourier integral operators.

Extension of estimates to nonlinear wave equations with rough initial data.

Applicability to compact Riemannian manifolds with bounded geometry.

Abstract

We introduce the Hardy spaces for Fourier integral operators on Riemannian manifolds with bounded geometry. We then use these spaces to obtain improved local smoothing estimates for Fourier integral operators satisfying the cinematic curvature condition, and for wave equations on compact manifolds. The estimates are essentially sharp, for all $2<p<\infty$ and on each compact manifold. We also apply our local smoothing estimates to nonlinear wave equations with initial data outside of $L^{2}$-based Sobolev spaces.