Damping oscillatory Integrals of convex analytic functions

TL;DR

This paper establishes optimal decay estimates for damping oscillatory integrals over convex analytic hypersurfaces in dimensions 2 and 3, with implications for restriction and convolution operators, advancing understanding of oscillatory integral behavior.

Contribution

It provides the first sharp decay bounds for these integrals in low dimensions and extends results to complex damping factors, with applications to harmonic analysis operators.

Findings

Optimal decay estimates for $( abla^{1/2} ext{measure})^rown$ in dimensions 2 and 3.

Extension of decay estimates to complex damping factors with polynomial growth in $|t|$.

Improved restriction estimates for convex analytic hypersurfaces in low dimensions.

Abstract

Let $H\subset \R^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $\sigma $ on $H$. Let $\kappa$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $(\kappa^{1/2} \sigma)^\wedge$ with the damping factor $\kappa^{1/2}$ and prove the optimal decay estimate \[ |(\kappa^{1/2} \sigma )^\wedge(\xi)|\le C|\xi|^{-d/2}\] 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--M\"uller. Furthermore, we prove the same estimates for $(\kappa^{1/2+it} \sigma )^\wedge$ 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 $\kappa^{1/(d+2)} \sigma$. This work was inspired by the stationary set method due to Basu--Guo--Zhang--Zorin-Kranich.