Estimates for capacities of nodal sets and polarity criteria in recurrent Dirichlet spaces

In this work we consider an irreducible and recurrent symmetric Dirichlet s pace (E, F) on a locally compact separable metric space X with reference me asure m, such that the transition function p(t)(x, .) of the associated Hun t's process is absolutely continuous w.r.t. m, for quasi every x is an elem ent of X. We then give estimates for (1-order) capacities Cap(N-upsilon) of the nodal set N-upsilon := {x is an element of X : upsilon(x) = 0} Of a fu nction upsilon in the extended space F-e, by proving Poincare-Wirtinger-typ e inequalities where the "constant" involved actually depends on Cap(N-upsi lon). We then derive estimates for the capacity of a closed set F, similar to those proved by K.-Th. Sturm. For example, when a Poincare-Wirtinger's i nequality holds true these estimates are of the following form: [GRAPHICS] where upsilon(F)(r) = m({x is an element of X : 0 < rho(x,F) < r}) is the v olume growth function of the Caratheodory metric rho associated with the st rongly local part of the Dirichlet space and R-F := sup(x is an element of X) rho(x, F). From these bounds criteria for polarity of not necessarily cl osed compact sets are also derived and examples are provided for diffusions in Euclidean domains with singular or degenerate coefficients.