Capacitary density and removable sets for Newton-Sobolev functions in metric spaces

TL;DR

This paper establishes that in certain metric spaces, sets meeting a capacitary density condition can be removed without affecting Newton-Sobolev functions, advancing understanding of function behavior in these spaces.

Contribution

It introduces a capacitary density condition that characterizes removable sets for Newton-Sobolev functions in metric spaces with doubling measures and Poincaré inequalities.

Findings

Sets satisfying the capacitary density condition are removable for Newton-Sobolev functions

The result applies to complete metric spaces with doubling measures and supporting a (1,1)-Poincaré inequality

Provides a new criterion for removability in analysis on metric spaces

Abstract

In a complete metric space equipped with a doubling measure and supporting a $(1,1)$-Poincar\'e inequality, we show that every set satisfying a suitable capacitary density condition is removable for Newton-Sobolev functions.