Sobolev spaces and hyperbolic fillings

TL;DR

This paper introduces a new fractional Sobolev space on Ahlfors regular metric spaces using hyperbolic fillings, connecting it to Haj ext{l}asz-Sobolev spaces under Poincaré inequality conditions.

Contribution

It defines a novel Sobolev space $A^p(Z)$ via hyperbolic fillings and establishes its equivalence with Haj ext{l}asz-Sobolev spaces when a Poincaré inequality holds.

Findings

$A^p(Z)$ is a new fractional Sobolev space for metric spaces.

$A^{Q}(Z)$ coincides with Haj ext{l}asz-Sobolev space under Poincaré inequality.

The space captures fractional smoothness via hyperbolic filling extensions.

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