Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties.

The basic object for the sequel is a fixed regul r Dirichlet form S with domain Q) ( } on a real Hubert space H = L(X, m). The underlying topological space X is a locally compact separable Hausdorff space and m is a positive Radon measure with supp[w] = X. The form S is always assumed to be strongly local (i.e. S (u, v) = 0 whenever u e Q) ((?) is constant on a neighborhood of the support of v €&(£)) and to be irreducible (i.e. we ^loc(<f) is constant on X whenever S (u, u) = 0). In other words, S has no killing measure and no jumping measure and X cannot be decomposed into (non-trivial) subsets which are invariant for S. For notions concerning Dirichlet forms we recommend the monograph [F] of M. Fukushima whose terminology we mostly follow. For a brief discussion of the notion or irreducibility we refer to the Appendix at the end of this article. Let us mention one crucial consequence of the regularity of S. Each function u€^(S) admits a quasi-continuous version (which is determined pointwise up to exceptional sets). For simplicity, we always write again u for and make the convention that whenever we use a pointwise version of u (e. g. in expressions like J φ or J M φ with a measure μ charging no exceptional sets) then {u>0} without restriction this version is always chosen quasi-continuous.