Higher rank spectral projector estimates and a Stein--Tomas restriction theorem

Source abstract

Let $X$ be a compact locally symmetric space of noncompact type. We prove sharp upper bounds for the $L^2\to L^p$ operator norms of spectral projectors for joint eigenfunctions of the full ring of invariant differential operators on $X$. This improves the corresponding result of Marshall, by removing a logarithmic power loss at the critical exponent. Using the same method, we also prove the endpoint case of Mockenhaupt's $(L^2,L^p)$ Fourier restriction theorem.