Pierre Germain (Imperial College)

L^p bounds for spectral projectors on Riemannian manifolds: the case of small bandwidth

Abstract:

Consider the Laplace-Beltrami operator on a Riemannian manifold. Through functional calculus, one defines spectral projectors at a given frequency, with a given bandwidth. What is the L² → L^p operator norm of these projectors? Sogge proved 40 years ago universal and optimal bounds if the bandwidth is sufficiently broad. For narrower bandwidth, the global geometry of the manifold comes into play, and the question is mostly open. I will present progress on these questions for some locally symmetric spaces: the Euclidean torus, and hyperbolic surfaces. This is joint work with J.-P. Anker, T. Leger and S. Myerson.