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 $X_{\epsilon }$. 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 $X_{\epsilon }$, and show that ${\mu }_{\beta }$ is doubling and supports a $1$-Poincaré 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 $B_{p,p}^{\theta }(Z)$ is the trace space of the Newton--Sobolev space $N^{1,p}(X_{\epsilon },{\mu }_{\beta })$ when $\epsilon =\log \alpha$. Finally, we exploit the tools of potential theory on $X_{\epsilon }$ to obtain fine properties of functions in $B_{p,p}^{\theta }(Z)$, 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 ${\mathbb{R}}^n$.

53 pages

References in corpus (2)

• Uniformizing Gromov hyperbolic spaces with Busemann functions
• Rough isometry between Gromov hyperbolic spaces and uniformization

Cited by in corpus (8)

• Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets
• Uniformizing Gromov hyperbolic spaces with Busemann functions
• Extension and trace theorems for noncompact doubling spaces
• Sharp Besov capacity estimates for annuli in metric spaces with doubling measures
• Embedded trace operator for infinite metric trees
• Uniformization, $\partial$-biLipschitz maps, sphericalization, and inversion
• Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted $L^q$ spaces on metric spaces
• Preserving Besov (fractional Sobolev) energies under sphericalization and flattening