$L^p-L^2$ Fourier restriction for hypersurfaces in $\Bbb R^3$: Part I

TL;DR

This paper establishes a sharp $L^p-L^2$ Fourier restriction theorem for a broad class of smooth, finite type hypersurfaces in three-dimensional space, including all real-analytic hypersurfaces, advancing understanding in harmonic analysis.

Contribution

It proves a sharp Fourier restriction theorem for hypersurfaces in $R^3$, extending previous results to a larger class of smooth and real-analytic hypersurfaces.

Findings

Proved a sharp $L^p-L^2$ restriction estimate for hypersurfaces.

Included all real-analytic hypersurfaces in the class.

Provided a modified, extended version ensuring compatibility with subsequent work.

Abstract

This is the first of two articles in which we prove a sharp $L^p-L^2$ Fourier restriction theorem for a large class of smooth, finite type hypersurfaces in $\Bbb R^3$, which includes in particular all real-analytic hypersurfaces. The present file is a modified and extended version of an earlier file of the same title. Some changes and corrections had become necessary, in order to make sure that Part II is really compatible with Part I, and it is recommended to replace the earlier version of Part I by the new one.