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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 $\cal A$.