Sobolev Algebras on Lie Groups and Riemannian Manifolds

We prove that on any connected unimodular Lie group G, the space$L_{\alpha}^{p}(G)\cap L^{\infty}(G)$, where$L_{\alpha}^{p}(G)$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.