Removable singularities for bounded $\mathcal{A}$-(super)harmonic and quasi(super)harmonic functions on weighted ${\mathbf{R}}^n$

Abstract

It is well known that sets of $p$-capacity zero are removable for bounded $p$-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted ${\mathbf{R}}^n$ when $n>p$, although only in very special cases. A complete characterization of removable singularities for bounded $\mathcal{A}$-harmonic functions on weighted ${\mathbf{R}}^n$, $n\ge 1$, is also given, where the weight is $p$-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted ${\mathbf{R}}^n$, $n\ge 2$, as well as on unweighted $\mathbf{R}$. For bounded $\mathcal{A}$-superharmonic functions and bounded quasisuperharmonic functions on weighted ${\mathbf{R}}^n$, $n\ge 2$, we show that relatively closed sets are removable if and only if they have zero capacity.