Séance Séminaire

Séminaire Gaston Darboux

L^p-positivity preservation on Riemannian manifolds

A Riemannian manifold is said to be L^p-positivity preserving if any L^p (distributional) solution of - Δu +u ≥ 0 is necessarily nonnegative. This property stems from the work of T. Kato and is motivated by, and has applications to, the spectral theory of Schrödinger operators with singular potentials. We will first show that complete smooth manifolds are always L^p-positivity preserving for finite p>1, solving in particular a conjecture by Braverman, Milatovic and Shubin. Then we will move some steps towards both incomplete manifolds and nonsmooth metric spaces. This is based on joint works with B. Guneysu, S. Pigola, P. Stollmann and D. Valtorta.