Sobolev space properties of superharmonic functions on metric spaces
In the recent years, after the publication of the important paper by \textit{J. Heinonen} and \textit{P. Koskela} [Acta Math. 181, No. 1, 1--61 (1998; Zbl 0915.30018)], there has been a period of intensive study of nonlinear analysis in metric spaces. The leading idea is to generalize, as much as possible, nonlinear potential theory and quasiconformal mapping theory to the setup of spaces that are not even locally euclidean, however, under some local homogeneity postulates such as the existence of a doubling Borel measure or the validity of some integral inequalities. This development of the past few years shows that starting with a few postulates much of the classical theory of superharmonic functions extends to this general setup. The authors prove a local integrability result for superharmonic functions generalizing a theorem of \textit{P. Lindqvist} [J. Reine Angew. Math. 365, 67--79 (1986; Zbl 0572.31004)].
scientific article; zbMATH DE number 2062006