LIPSCHITZ-SPACES AND POINCARE INEQUALITIE S

In the setting of infinite graphs and non-compact Riemannian manifolds , we show that suitable families of Poincare inequalities yield global embeddings of Sobolev spaces into Lipschitz spaces, as well as Trudin ger type inequalities. This applies for example to cocompact coverings and to manifolds that are roughly isometric to a manifold with nonneg ative 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 interpretatio n of some of our results in terms of distances on graphs associated wi th the L(p) norm of the gradient. (C) 1996 Academic Press, Inc.