Oscillatory integral operators with homogeneous phase functions

TL;DR

This paper establishes sharp $L^p$ estimates for oscillatory integral operators with homogeneous phase functions, extending Fourier extension results and applying them to wave equations on manifolds.

Contribution

It extends $L^p$ estimates for oscillatory integrals with homogeneous phases using polynomial partitioning and applies these results to local smoothing for wave equations on manifolds.

Findings

Proved $L^p$ estimates for cone Fourier extension operators with homogeneous phases.

Provided examples demonstrating the sharpness of these estimates.

Derived new local smoothing estimates for wave equations on compact Riemannian manifolds.

Abstract

Oscillatory integral operators with $1$-homogeneous phase functions satisfying a convexity condition are considered. For these we show the $L^p - L^p$-estimates for the Fourier extension operator of the cone due to Ou--Wang via polynomial partitioning. For this purpose, we combine the arguments of Ou--Wang with the analysis of Guth--Hickman--Iliopoulou, who previously showed sharp $L^p-L^p$-estimates for non-homogeneous phase functions with variable coefficients under a convexity assumption. Furthermore, we provide examples exhibiting Kakeya compression, which shows the estimates to be sharp. We apply the oscillatory integral estimates to show new local smoothing estimates for wave equations on compact Riemannian manifolds $(M,g)$ with $\dim M \geq 3$. This generalizes the argument for the Euclidean wave equation due to Gao--Liu--Miao--Xi.