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