Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets

TL;DR

This paper characterizes removable sets for Newtonian Sobolev functions in metric spaces, extending known results to the case p=1 and exploring properties of p-path almost open sets, including their measurability and topological features.

Contribution

It provides a new characterization of removable sets for Sobolev spaces at p=1 and introduces a detailed analysis of p-path almost open sets in metric measure spaces.

Findings

A closed measure-zero set is removable iff its complement supports a p-Poincaré inequality.

Extension of Sobolev functions from noncomplete to complete spaces for p=1.

Existence of nonmeasurable p-path almost open sets under the continuum hypothesis.

Abstract

We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincar\'e inequality. In particular, when restricted to Euclidean spaces, a closed set $E\subset \mathbf{R}^n$ with zero Lebesgue measure is shown to be removable for $W^{1,p}(\mathbf{R}^n \setminus E)$ if and only if $\mathbf{R}^n \setminus E$ supports a $p$-Poincar\'e inequality as a metric space. When $p>1$, this recovers Koskela's result (Ark. Mat. 37 (1999), 291--304), but for $p=1$, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces $L^{1,p}$. To be able to include $p=1$, we first study extensions of Newtonian Sobolev functions in the case $p=1$ from a noncomplete space $X$ to its completion $\widehat{X}$. In these results, $p$-path almost open sets play an important role, and we provide a characterization of them by means of $p$-path open, $p$-quasiopen and $p$-finely open sets. We also show that there are nonmeasurable $p$-path almost open subsets of $\mathbf{R}^n$, $n \geq 2$, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with $L^p$-integrable upper gradients, about $p$-quasiopen, $p$-path and $p$-finely open sets, and about Lebesgue points for $N^{1,1}$-functions, to spaces that only satisfy local assumptions.