Espaces de Lipschitz et inégalités de Poincaré

In the setting of infinite graphs and non-compact Riemannian manifolds, we show that suitable families of Poincaré inequalities yield global embeddings of Sobolev spaces into Lipschitz spaces, as well as Trudinger type inequalities. This applies for example to cocompact coverings and to manifolds that are roughly isometric to a manifold with nonnegative Ricci curvature. In the process, we give several reformulations of the Sobolev inequalities, and in particular show their equivalence with some L^p Faber-Krahn inequalities. We also give an interpretation of some of our results in terms of distances on graphs associated with the L^p norm of the gradient.