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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...

We develop a theory of removable singularities for the weighted Bergman space ${𝒜}_{\mu }^p\left(\Omega \right)=\{f\text{analytic}\text{in}\Omega {\int }_{\Omega }{\left|f\right|}^p\mathrm{d}\mu <\infty \}$, where $\mu$ is a Radon measure on $ℂ$. The set $A$ is weakly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)\subset \text{Hol}\left(\Omega \right)$, and strongly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)={𝒜}_{\mu }^p\left(\Omega \right)$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathrm{B}MO$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable....

Let $X$ be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré inequality. We show that open subsets of $X$ can be approximated by regular sets. This has applications in nonlinear potential theory on metric spaces. In particular it makes it possible to define Wiener solutions of the Dirichlet problem for $p$-harmonic functions and to show that they coincide with three other notions of generalized solutions.