Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
Anders Björn, Jana Björn, Nageswari Shanmugalingam
Abstract
In this paper we study connections between Besov spaces of functions on a compactmetric space Z, equipped with a doubling measure, and the Newton–Sobolev spaceof functions on a uniform domain Xε. This uniform domain is obtained as auniformization of a (Gromov) hyperbolic filling of Z. To do so, we construct afamily of hyperbolic fillings in the style of Bonk–Kleiner [9] and Bourdon–Pajot [13]. Then for each parameter β &gt; 0 we construct a lift μβ of the doubling measure νon Z to Xε, and show that μβ is doubling and supports a 1-Poincaré inequality.We then show that for each θ with 0 &lt; θ &lt; 1 and p ≥ 1 there is a choice of β = p(1 − θ)ε such that the Besov space <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?B%5E%7B%5Ctheta%7D_%7Bp,p%7D(Z)" data-classname="equation" data-title=" /> is the trace space of the Newton–Sobolev space N1,p(Xε, μβ). Finally, we exploit the tools of potential theory on Xεto obtain fine properties of functions in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?B%5E%7B%5Ctheta%7D_%7Bp,p%7D(Z)" data-classname="equation" data-title=" />, such as their quasicontinuity andquasieverywhere 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 tocompact subsets of Euclidean spaces improves upon a result of Netrusov [43] in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctextbf%7BR%7D%5E%7Bn%7D" data-classname="equation" data-title=" />.