Closability of Dirichlet forms and Poincaré-type inequalities (Q1902486)

Let \(X\) be a locally compact space with countable basis, \(m\) a bounded measure on \(X\), and \(H \subset C_0 (X)\) a dense vector lattice stable for contractions. Let \(a\) be a quadratic form on \(H\) such that the contraction operates on \(a\). A positive measure \(\nu\) is said to satisfy the Poincaré inequality on an open subset \(\omega \subset X\) if \(\int u^2 d \nu \leq Ca (u,u)\) for all \(u \in H\) with compact support contained in \(\omega\). In this paper the author studies the relation between the closability of the form \(a\) and the existence of sufficiently many measures \(\nu\) satisfying the Poincaré inequality on open subsets of \(X\).