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 Xε. 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 Bonk–Kleiner [9] and Bourdon–Pajot [13]. Then for each parameter β>0 we construct a lift μβ of the doubling measure ν on Z to Xε, and show that μβ is doubling and supports a 1-Poincaré inequality. We then show that for each θ with 0<θ<1 and p≥1 there is a choice of β=p(1−θ)ε such that the Besov space Bp,pθ(Z) is the trace space of the Newton–Sobolev space N1,p(Xε,μβ). Finally, we exploit the tools of potential theory on Xε to obtain fine properties of functions in Bp,pθ(Z), such as their quasicontinuity and quasieverywhere existence of Lq-Lebesgue points with q=sνp/(sν−pθ), where sν 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 [43] in Rn.
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