Damping oscillatory Integrals of convex analytic functions

Abstract

Let $H\subset {\mathbb{R}}^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $\ddot{I}ƒ$ on $H$. Let $\hat{I}º$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $(\hat{I}{º}^{1/2}\ddot{I}ƒ{)}^{\wedge }$ with the damping factor $\hat{I}{º}^{1/2}$ and prove the optimal decay estimate \[ |(κ^{1/2} σ)^\wedge(ξ)|\le C|ξ|^{-d/2}\] for $d=2,3,$ and with an extra logarithmic factor for $d=4$. Our result provides an essentially complete answer, since such decay estimates generally fail for $d\ge 5$, even for convex analytic hypersurfaces, as shown by Cowling--Disney--Mauceri--Müller. Furthermore, we prove the same estimates for $(\hat{I}{º}^{1/2+it}\ddot{I}ƒ{)}^{\wedge }$ with $C$ growing polynomially in $|t|$. As consequences, we obtain the best possible estimates for the convolution, maximal, and adjoint restriction operators associated with $H$, incorporating the mitigating factors of optimal orders. In particular, for $d=2,3$, we prove the $L^2$--$L^{2(d+2)/(d+4)}$ restriction estimate with respect to the affine surface measure $\hat{I}{º}^{1/(d+2)}\ddot{I}ƒ$. This work was inspired by the stationary set method due to Basu--Guo--Zhang--Zorin-Kranich.

The references have been updated, along with slight modifications to the abstract and introduction