Limits at infinity for Haj{\l}asz-Sobolev functions in metric spaces

TL;DR

This paper investigates the behavior at infinity of Haj{ }asz-Sobolev functions in metric spaces, establishing conditions under which these functions have well-defined limits outside negligible sets.

Contribution

It introduces a new framework using variational relative capacity to analyze limits at infinity, extending prior results for Newtonian and fractional Sobolev functions.

Findings

Quasicontinuous representatives have limits at infinity outside exceptional sets

The approach refines earlier methods based on Hausdorff content

Results extend to homogeneous Newtonian and fractional Sobolev functions

Abstract

We study limits at infinity for homogeneous Hajlasz-Sobolev functions defined on uniformly perfect metric spaces equipped with a doubling measure. We prove that a quasicontinuous representative of such a function has a pointwise limit at infinity outside an exceptional set, defined in terms of a variational relative capacity. Our framework refines earlier approaches that relied on Hausdorff content rather than relative capacity, and it extends previous results for homogeneous Newtonian and fractional Sobolev functions.