Oscillatory integral operators with homogeneous phase functions

Abstract

AbstractOscillatory integral operators with 1-homogeneous phase functions satisfying a convexity condition are considered. For these we show the Lp–Lp-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 Lp–Lp-estimates for non-homogeneous phase functions with variable coefficients under a convexity assumption. Furthermore, we provide examples exhibiting Kakeya compression, which shows a more restrictive range than dictated by the Knapp example in higher dimensions. We apply the oscillatory integral estimates to show new local smoothing estimates for wave equations on compact Riemannian manifolds (M, g) with dim M ≥ 3. This generalizes the argument for the Euclidean wave equation due to Gao–Liu–Miao–Xi.