Sobolev algebras on Lie groups and Riemannian manifolds

We prove that on any connected unimodular Lie group G, the space L^p_α(G) ∩ L^∞(G), where L^p_α(G) is the Sobolev space of order α > 0 associated with a sublaplacian, is an algebra under pointwise product. This generalizes results due to Strichartz (in the Euclidean case), to Bohnke (in the case of stratified groups), and others. A global version of this fact holds for groups with polynomial growth. We give similar results for Riemannian manifolds with Ricci curvature bounded from below, respectively nonnegative.