An affine Fourier restriction theorem for conical surfaces

TL;DR

This paper establishes an affine-invariant Fourier restriction estimate for conical surfaces using weighted measures, leading to sharp $L^p-L^q$ restriction results and discussing optimality and anomalies in existing theories.

Contribution

It introduces a new affine-invariant restriction theorem for conical surfaces with optimal weights, extending previous results and addressing anomalies in the literature.

Findings

Proves a Fourier restriction estimate with a weighted measure for conic surfaces.

Demonstrates the sharpness of the restriction theorem up to an endpoint.

Provides a discussion on anomalies and type k conical restriction theorems.

Abstract

A Fourier restriction estimate is obtained for a broad class of conic surfaces by adding a weight to the usual underlying measure. The new restriction estimate exhibits a certain affine-invariance and implies the sharp $L^p-L^q$ restriction theorem for compact subsets of a type $k$ conical surface, up to an endpoint. Furthermore, the chosen weight is shown to be, in some quantitative sense, optimal. Appended is a discussion of type k conical restriction theorems which addresses some anomalies present in the existing literature.