Extension and trace results for doubling metric measure spaces and their hyperbolic fillings

TL;DR

This paper establishes a connection between Besov spaces on a compact metric space and Newton–Sobolev spaces on hyperbolic fillings, providing new insights into function properties and trace characterizations in metric measure spaces.

Contribution

It constructs hyperbolic fillings and measures to relate Besov and Newton–Sobolev spaces, extending potential theory tools to analyze function properties on these spaces.

Findings

Besov space $B^ heta_{p,p}(Z)$ is the trace of $N^{1,p}(X_eta, u_eta)$

Constructed measures support Poincaré inequalities and are doubling

Improved regularity results for functions on Euclidean subsets

Abstract

In this paper we study connections between Besov spaces of functions on a compact metric space $Z$, equipped with a doubling measure, and the Newton--Sobolev space of functions on a uniform domain $X_\varepsilon$. This uniform domain is obtained as a uniformization of a (Gromov) hyperbolic filling of $Z$. To do so, we construct a family of hyperbolic fillings in the style of the work of Bonk and Kleiner and the work of Bourdon and Pajot. Then for each parameter $\beta>0$ we construct a lift $\mu_\beta$ of the doubling measure $\nu$ on $Z$ to $X_\varepsilon$, and show that $\mu_\beta$ is doubling and supports a $1$-Poincar\'e inequality. We then show that for each $\theta$ with $0<\theta<1$ and $p\ge 1$ there is a choice of $\beta=p(1-\theta)\log\alpha$ such that the Besov space $B^\theta_{p,p}(Z)$ is the trace space of the Newton--Sobolev space $N^{1,p}(X_\varepsilon,\mu_\beta)$ when $\varepsilon=\log\alpha$. Finally, we exploit the tools of potential theory on $X_\varepsilon$ to obtain fine properties of functions in $B^\theta_{p,p}(Z)$, such as their quasicontinuity and quasieverywhere existence of $L^q$-Lebesgue points with $q=s_\nu p/(s_\nu-p\theta)$, where $s_\nu$ is a doubling dimension associated with the measure $\nu$ on $Z$. Applying this to compact subsets of Euclidean spaces improves upon a result of Netrusov in $\mathbb{R}^n$.