Characterizations of $p$-superharmonic functions on metric spaces

Volume 169 / 2005

Anders Björn Studia Mathematica 169 (2005), 45-62 MSC: Primary 31C45; Secondary 31C05, 35J60, 49J27. DOI: 10.4064/sm169-1-3

Abstract

We show the equivalence of some different definitions of $p$-superharmonic functions given in the literature. We also provide several other characterizations of $p$-superharmonicity. This is done in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. There are many examples of such spaces. A new one given here is the union of a line (with the one-dimensional Lebesgue measure) and a triangle (with a two-dimensional weighted Lebesgue measure). Our results also apply to Cheeger $p$-superharmonic functions and in the Euclidean setting to $\cal A$-superharmonic functions, with the usual assumptions on $\cal A$.

Authors

  • Anders BjörnDepartment of Mathematics
    Linköpings universitet
    SE-581 83 Linköping, Sweden
    e-mail

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