Capacitary criteria for Poincaré-type inequalities (Q1781892)

Let \(E\) be a locally compact separable metric space and let \((D, {\mathcal D}(D))\) be regular Dirichlet form on a normed linear space \((\mathbf B, \| .\|_{\mathbf B},\mu)\) of real functions on \(E\). Under some natural assumptions on \(\mathbf B\), the author estimates the optimal constant \(A_{\mathbf B}\) in the Poincaré-type inequality (\(\| f^2\|_{\mathbf B}\leq A_{\mathbf B} D(f); f\in {\mathcal D}(D)\cap C_0(E)\)). As in the classical situation, the lower and upper bounds of \(A_{\mathbf B}\) are given, for the transient case and for the recurrent one, in terms of the capacity. This result may be applied for Banach or Orlicz spaces. Moreover, explicit estimations are computed in dimension one.