Oscillatory integral operators with homogeneous phase functions

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 $M\ge3$.