Existence and uniqueness of limits at infinity for homogeneous Sobolev functions

TL;DR

This paper proves the existence and uniqueness of limits at infinity for homogeneous Sobolev functions on metric measure spaces with doubling measures and Poincaré inequalities, introducing weak polar coordinates and radial limits.

Contribution

It establishes conditions for limits at infinity in Sobolev spaces on metric measure spaces and introduces new notions of weak polar coordinates and radial curves.

Findings

Limits at infinity are unique outside zero modulus families.

Conditions for the existence of radial limits are characterized.

Applications to specific metric measure space settings.

Abstract

We establish the existence and uniqueness of limits at infinity along infinite curves outside a zero modulus family for functions in a homogeneous Sobolev space under the assumption that the underlying space is equipped with a doubling measure which supports a Poincar\'e inequality. We also characterize the settings where this conclusion is nontrivial. Secondly, we introduce notions of weak polar coordinate systems and radial curves on metric measure spaces. Then sufficient and necessary conditions for existence of radial limits are given. As a consequence, we characterize the existence of radial limits in certain concrete settings.