A Fourier Restriction Theorem For A Twodimensional Surface Of Finite Type

TL;DR

This paper investigates Fourier restriction estimates for a broad class of two-dimensional surfaces with varying curvature in R^3, extending known results through a bilinear method adaptation.

Contribution

It introduces new Fourier restriction estimates for surfaces of finite type with variable curvature, expanding the scope beyond analytic hypersurfaces.

Findings

Established new L^p to L^q restriction estimates for surfaces with finite type.

Developed an adapted bilinear method for surfaces with strongly varying curvature.

Identified novel features in the restriction problem for non-analytic hypersurfaces.

Abstract

The problem of $L^p(R^3)\to L^2(S)$ Fourier restriction estimates for smooth hypersurfaces S of finite type in R^3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up the study of more general $L^p(R^3)\to L^q(S)$ Fourier restriction estimates, by studying a prototypical class of two-dimensional surfaces with strongly varying curvature conditions. Our approach is based on an adaptation of the so-called bilinear method. We discuss several new features arising in the study of this problem.