Analysis on local Dirichlet spaces. I: Recurrence, conservativeness and \(L^ p\)-Liouville properties. (English) Zbl 0806.53041

Local Dirichlet spaces are used as an appropriate frame to unify and extend various famous results from differential geometry obtained originally by Yau, Cheng/Yau, Karp, Karp/Li, Grigor’yan. Sharp conditions for recurrence as well as for conservativeness and for exponential instability are given. These conditions are in terms of the volume growth \(r \mapsto m(B_ r(x))\) of metric balls \(B_ r(x) \subset X\) which are defined intrinsically by the respective Dirichlet form on \(L^ 2(X,m)\). Also \(L^ p\)-growth conditions for nonnegative sub- or supersolutions on \(X\) are derived. In particular, \(L^ p\)-Liouville theorems are obtained.


31C25 Dirichlet forms
60J45 Probabilistic potential theory
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
Full Text: DOI Crelle EuDML