Endpoint bounds for a class of spectral multipliers on compact manifolds

TL;DR

This paper refines endpoint bounds for spectral multipliers on compact manifolds, extending previous results by incorporating Besov space estimates and building on recent advances in radial Fourier multiplier analysis.

Contribution

It provides improved endpoint bounds for spectral multipliers using Besov space techniques, enhancing prior spectral projection estimates on compact manifolds.

Findings

Refined endpoint bounds for spectral multipliers in Besov spaces

Extension of Seeger's endpoint results with new techniques

Application of recent radial Fourier multiplier methods

Abstract

It is well known that the Stein-Tomas $L^2$ Fourier restriction theorem can be used to derive sharp $L^p$ bounds for radial Fourier multipliers such as the Bochner-Riesz means. In a similar manner, $L^p \to L^2$ estimates for spectral projection operators have been utilized in order to obtain sharp $L^p$ bounds for spectral multipliers of self-adjoint elliptic pseudo-differential operators on compact manifolds. In this paper, we refine an endpoint result for spectral multipliers due to Seeger, providing endpoint bounds in terms of Besov spaces. Our proof is based on the ideas from the recent work by Heo, Nazarov and Seeger, and Lee, Rogers and Seeger on radial Fourier multipliers.