Stein-Tomas Restriction Theorem via Spectral Measure on Metric Measure Spaces

TL;DR

This paper provides a complete proof of the Stein-Tomas restriction theorem using spectral measures on metric measure spaces, extending previous partial results and applying to asymptotically conic manifolds.

Contribution

It offers a full proof of the spectral measure-based restriction theorem on general metric measure spaces, broadening the scope beyond special cases.

Findings

Complete proof of the spectral measure restriction theorem

Extension to asymptotically conic manifolds

Generalization to metric measure spaces

Abstract

The Stein-Tomas restriction theorem on Euclidean space says one can meaningfully restrict $\hat{f}$ to the unit sphere of $\mathbb{R}^n$ provided $f \in L^p(\mathbb{R}^n)$ with $1 < p < 2$. This result can be rewritten in terms of the estimates for the spectral measure of Laplacian. Guillarmou, Hassell and Sikora formulated a sufficient condition of the restriction theorem, via spectral measure, on abstract metric measure spaces. But they only proved the result in a special case. The present note aims to give a complete proof. In the end, it will be applied to the restriction theorem on asymptotically conic manifolds.