Oscillatory integral operators with homogeneous phase functions

Robert Schippa

Published 2021-09-28, updated 2021-11-18Version 2

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. The estimates are supplemented by examples exhibiting Kakeya compression. We apply the estimates to show new local smoothing estimates for wave equations on compact Riemannian manifolds $(M,g)$ with $\dim M \geq 3$.