On Removable Sets for Weighted Sobolev Functions

TL;DR

This paper establishes geometric conditions under which compact null sets are removable for weighted Sobolev functions, using cube coverings and analysis on metric spaces, extending beyond classical weight classes.

Contribution

It introduces new porosity conditions for removability that do not rely on capacity, applicable to a broad class of doubling weights satisfying Poincaré inequalities.

Findings

Provides geometric criteria for removability of null sets

Extends results to weights beyond Muckenhoupt class

Uses metric space analysis techniques

Abstract

We give sufficient geometric conditions, not involving capacities, for a compact null set to be removable for the Sobolev functions on weighted $\mathbb R^n$, defined as the closure of smooth functions in the weighted Sobolev norm. Our porosity conditions are in terms of suitable coverings by cubes. The weights are assumed to be doubling and satisfy a Poincar\'e inequality, which includes, but is not equal to, the famous class of Muckenhoupt weights. Our proofs use ideas and techniques from the theory of analysis on metric spaces.