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-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 Bonk-Kleiner [9] and Bourdon-Pajot [13]. 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-Poincare inequality. We then show that for each theta with 0 < theta < 1 and p >= 1 there is a choice of beta = p(1 - theta)epsilon such that the Besov space B-p,p(theta)(Z) is the trace space of the Newton-Sobolev space N-1,N-P(X-epsilon, mu(beta)). Finally, we exploit the tools of potential theory on X-epsilon to obtain fine properties of functions in beta(theta)(p,p) (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 [43 ]in R-n. (C) 2021 The Author(s). Published by Elsevier Masson SAS.

Automated summary

This paper studies the connections between Besov spaces of functions on a compact metric space and the Newton-Sobolev space of functions on a uniform domain. By constructing hyperbolic fillings and lifting measures, the trace space equality between the two function spaces is obtained. Finally, fine properties of functions are obtained using potential theory tools.