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

TL;DR

This paper characterizes removable singularities for bounded -harmonic and quasiharmonic functions on weighted  spaces, revealing conditions under which sets of positive capacity are removable.

Contribution

It provides a complete characterization of removable singularities for bounded -harmonic functions on weighted  spaces, extending classical results to weighted and unweighted cases.

Findings

Removable sets of positive capacity can occur on unweighted  when n > p.

Zero capacity is necessary and sufficient for removability of relatively closed sets for superharmonic functions.

Characterization applies to both -harmonic and quasiharmonic functions on weighted and unweighted  spaces.

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.