Characterizations for the existence of traces of first-order Sobolev spaces on hyperbolic fillings

TL;DR

This paper provides characterizations for when traces of first-order Sobolev spaces exist on hyperbolic fillings of compact metric spaces with doubling measures, linking boundary properties to Sobolev trace existence.

Contribution

It introduces new criteria for the existence of Sobolev traces on hyperbolic fillings, connecting geometric boundary conditions with functional space properties.

Findings

Characterizations for Sobolev trace existence on hyperbolic fillings

Conditions linking boundary measure and Sobolev spaces

Framework applicable to compact metric spaces with doubling measures

Abstract

In this paper, we study the existence of traces for Sobolev spaces on the hyperbolic filling $X$ of a compact metric space $Z$ equipped with a doubling measure. Given a suitable metric on $X$, we can regard $Z$ as the boundary of $X$. After equipping $X$ with a weighted measure $\mu$ via the measure on $Z$ and the Euclidean arc length, we give characterizations for the existence of traces for first-order Sobolev spaces.