Sobolev Extension Domains on Metric Spaces of Homogeneous Type

Abstract

Let $(X,d,\mu)$ be a metric measure space of homogeneous type with a finite measure. Assume that $\Omega \subset X$ is a bounded domain, which satisfies an $A^*(\varepsilon,\delta)$-condition and $\mu(\partial\Omega)=0$. We show that there exists a bounded linear extension operator $Ext$ from the Hajłasz space $M^{1,p}(\Omega,d,\mu)$ to $M^{1,p}(X,d,\mu)$, such that $Ext (u)\vert_\Omega = u$.