Extensions of the Stein-Tomas theorem

TL;DR

This paper advances the Stein-Tomas restriction theorem by establishing an endpoint version with improved Lorentz space estimates, also extending similar enhancements to related oscillatory integral estimates and spectral projections.

Contribution

It introduces an endpoint version of the Stein-Tomas theorem for general measures with strengthened Lorentz space bounds, and applies these improvements to related oscillatory integral operators.

Findings

Proved an endpoint Stein-Tomas restriction theorem with generalized measures.

Achieved improved Lorentz space estimates for spectral and oscillatory integral operators.

Extended results to operators with fold singularities on compact manifolds.

Abstract

We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein's estimate on oscillatory integrals of Carleson-Sj\"olin-H\"ormander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with one-sided fold singularities.