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

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 $β>0$ we construct a lift $μ_β$ of the doubling measure $ν$ on $Z$ to $X_\varepsilon$, and show that $μ_β$ is doubling and supports a $1$-Poincaré inequality. We then show that for each $θ$ with $0<θ<1$ and $p\ge 1$ there is a choice of $β=p(1-θ)\logα$ such that the Besov space $B^θ_{p,p}(Z)$ is the trace space of the Newton--Sobolev space $N^{1,p}(X_\varepsilon,μ_β)$ when $\varepsilon=\logα$. Finally, we exploit the tools of potential theory on $X_\varepsilon$ to obtain fine properties of functions in $B^θ_{p,p}(Z)$, such as their quasicontinuity and quasieverywhere existence of $L^q$-Lebesgue points with $q=s_νp/(s_ν-pθ)$, where $s_ν$ is a doubling dimension associated with the measure $ν$ on $Z$. Applying this to compact subsets of Euclidean spaces improves upon a result of Netrusov in $\mathbb{R}^n$.

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