Square function estimates and Local smoothing for Fourier Integral Operators

TL;DR

This paper establishes a variable coefficient square function estimate that leads to sharp local smoothing results for Fourier integral operators in 2+1 dimensions, resolving the local smoothing conjecture for wave equations on compact surfaces.

Contribution

It introduces a variable coefficient square function estimate and applies it to prove the full range of sharp local smoothing estimates, settling a longstanding conjecture.

Findings

Proves a variable coefficient square function estimate.

Establishes sharp local smoothing estimates for 2+1 dimensional Fourier integral operators.

Completes the proof of the local smoothing conjecture for wave equations on compact Riemannian surfaces.

Abstract

We prove a variable coefficient version of the square function estimate of Guth--Wang--Zhang. By a classical argument of Mockenhaupt--Seeger--Sogge, it implies the full range of sharp local smoothing estimates for $2+1$ dimensional Fourier integral operators satisfying the cinematic curvature condition. In particular, the local smoothing conjecture for wave equations on compact Riemannian surfaces is completely settled.