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

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 $\mathbb{R}^n$.