Criteria to the Poincaré inequality associated with Dirichlet forms in ℝd, d ≥ 2

Abstract

We study the compactness of the resolvent of Witten Laplacians on 0-forms and more generally the validity of the Poincaré inequality for Dirichlet forms in finite dimension. We are interested in the case where some known sufficient condition becomes degenerate. For some classes of polyhomogeneous functions, we get simple conditions which are necessary and sufficient for the Poincaré inequality.