Extension and trace theorems for noncompact doubling spaces

TL;DR

This paper extends trace and extension theorems for Besov spaces to noncompact doubling metric spaces using a novel uniformization approach based on Busemann functions, leading to new regularity and embedding results.

Contribution

It introduces a new uniformization method for Gromov hyperbolic spaces that generalizes previous trace and extension theorems to noncompact settings, with applications to Besov spaces.

Findings

Existence of quasicontinuous representatives for Besov trace spaces

Almost everywhere Lebesgue points with respect to Besov capacity

Embeddings of Besov spaces into Hölder spaces

Abstract

We generalize the extension and trace results of Bj\"orn-Bj\"orn-Shanmugalingam \cite{BBS21} to the setting of complete noncompact doubling metric measure spaces and their uniformized hyperbolic fillings. This is done through a uniformization procedure introduced by the author that uniformizes a Gromov hyperbolic space using a Busemann function instead of the distance functions considered in the work of Bonk-Heinonen-Koskela \cite{BHK}. We deduce several corollaries for the Besov spaces that arise as trace spaces in this fashion, including the existence of representatives that are quasicontinuous with respect to the Besov capacity, the existence of $L^p$-Lebesgue points quasieverywhere with respect to the Besov capacity, embeddings into H\"older spaces for appropriate exponents, and a stronger Lebesgue point result under an additional reverse doubling hypothesis on the measure. We also obtain several Poincar\'e-type inequalities relating integrals of Besov functions over balls to integrals of upper gradients of extension of these functions to a uniformized hyperbolic filling of the space.