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

Sturm, Karl-Theodor

J. Reine Angew. Math. 456, 173-196 (1994).

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.

Reviewer: K.-Theodor Sturm (Erlangen)

MSC:

31C25	Dirichlet forms
60J45	Probabilistic potential theory
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions

Keywords:

local Dirichlet spaces; recurrence; conservativeness; exponential instability; volume growth; Dirichlet form; \(L^ p\)-Liouville theorems