APDE Seminar@BCAM: Estimates for oscillatory integrals with damping factors

Abstract

Let $\sigma$ be the surface measure on a smooth hypersurface $\mathcal H \subset \mathbb{R}^{d+1}$. A fundamental subject in harmonic analysis is to determine the decay of $\widehat{\sigma}$. For nondegenerate $\mathcal H$, the stationary phase method yields the optimal decay, while sharp bounds in the degenerate case are known only in limited situations.  In this talk, we are concerned with  the oscillatory estimate

$$

|(\kappa^{1/2}\sigma)^\wedge(\xi)| \le C|\xi|^{-d/2},

$$

for convex analytic surfaces $\mathcal H$, where $\kappa$ is the Gaussian curvature.   The damping factor $\kappa^{1/2}$ is expected to  recover the optimal decay, as suggested by the stationary phase expansion, but the work of Cowling–Disney–Mauceri–Müller shows that such bounds fail in general for $d \ge 5$ even when the surface is convex and analytic. However, it has remained open whether the estimate holds in lower dimensions $2\le d\le 4$. We establish it for $d=2,3$, and with a logarithmic loss for $d=4$. Our approach is inspired by the stationary set method of Basu–Guo–Zhang–Zorin-Kranich. We also discuss applications to convolution, maximal, and adjoint restriction operators.  This talk is based on joint work with Sewook Oh.