Sobolev spaces and hyperbolic fillings

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 $Q$-Poincar\'e inequality with $Q>1$, then $A^{Q}(Z)$ coincides with the familiar (homogeneous) Haj\l asz-Sobolev space.