Lipschitz spaces and Poincaré inequalities. (Espaces de Lipschitz et inégalités de Poincaré.) (French) Zbl 0859.58009

Coulhon, Thierry

J. Funct. Anal. 136, No. 1, 81-113 (1996).

The author shows that in the setting of infinite graphs and non-compact Riemannian manifolds, the 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 this process, the author gives several reformulations of the Sobolev inequalities, and in particular shows their equivalence with some \(L^p\) Faber-Krahn inequalities. An interpretation of some of the obtained results is also given in terms of distances on graphs associated with the \(L^p\) norm of the gradient.

Reviewer: N.Papaghiuc (Iaşi)

MSC:

58E35

Variational inequalities (global problems) in infinite-dimensional spaces

Keywords:

Poincaré inequalities; Lipschitz spaces; Trudinger type inequalities; Sobolev inequalities; Faber-Krahn inequalities; \(L^ p\) norm