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

The Stein-Tomas restriction theorem on Euclidean space says one can meaningfully restrict $\widehat{f}$ 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.