Sobolev spaces and hyperbolic fillings

Abstract

Let $Z$ be an Ahlfors $Q$-regular compact metric measure space, where $Q>0$. For $p>1$ we introduce a new (fractional) Sobolev space $A^p(Z)$ consisting of functions whose extensions to the hyperbolic filling of $Z$ satisfies a weak-type gradient condition. If $Z$ supports a $Q$-Poincaré inequality with $Q>1$, then $A^Q(Z)$ coincides with the familiar (homogeneous) Hajłasz-Sobolev space.

31 pages. Revised version. To appear in J. Reine Angew. Math