On the measure of the boundary of a Hajłasz-Sobolev $(M^{1,p},M^{1,q})$-extension domain

Abstract

We prove that the boundary of every bounded $(M^{1,p},M^{1,q})$-extension domain in a $Q$-doubling metric measure space has measure zero whenever $0<q<p<\infty$, with the additional restriction that $p<qQ/(Q-q)$ if $q<Q$.