Title: Endpoint bounds for spectral multipliers on compact manifolds.
Abstract: As is well known, the Stein-Tomas Fourier restriction theorem has applications to radial Fourier multipliers, such as the Bochner-Riesz means. There are analogous (and more general) results for spectral multipliers of the Laplace-Beltrami operator on compact Riemannian manifolds. In particular, weak-type endpoint bounds for the Bochner-Riesz means in such a setting are known in the range $1\leq p \leq 2(d+1)/(d+3)$ by Christ-Sogge, Seeger, and Tao. We present an endpoint spectral multiplier theorem which applies to multipliers in (localized) Besov spaces. We also presents some results on the necessary and sufficient conditions for the $L^p(R^d)$ boundedness of maximal operators generated by a class of (quasi) radial Fourier multipliers for some $p>2$. The results were inspired by the recent work on radial Fourier multipliers by Heo-Nazarov-Seeger and Lee-Rogers-Seeger.