Extension and trace theorems for noncompact doubling spaces

Abstract

We generalize the extension and trace results of Björn-Björn-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 Hölder spaces for appropriate exponents, and a stronger Lebesgue point result under an additional reverse doubling hypothesis on the measure. We also obtain several Poincaré-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.

63 pages. v3: Extensive revisions with several new and refined results. Some content from the previous version has been moved to arXiv:2101.03092

References in corpus (6)

• Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
• Besov spaces via hyperbolic fillings
• Uniformizing Gromov hyperbolic spaces with Busemann functions
• Weak Capacity and Critical Exponents
• Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces
• Uniformization, $\partial$-biLipschitz maps, sphericalization, and inversion

Cited by in corpus (4)

• Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets
• Uniformizing Gromov hyperbolic spaces with Busemann functions
• Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces
• Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted $L^q$ spaces on metric spaces