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We generalize the extension and trace results ofBj\"orn-Bj\"orn-Shanmugalingam \cite{BBS21} to the setting of completenoncompact doubling metric measure spaces and their uniformized hyperbolicfillings. This is done through a uniformization procedure introduced by theauthor that uniformizes a Gromov hyperbolic space using a Busemann functioninstead of the distance functions considered in the work ofBonk-Heinonen-Koskela \cite{BHK}. We deduce several corollaries for the Besovspaces that arise as trace spaces in this fashion, including the existence ofrepresentatives that are quasicontinuous with respect to the Besov capacity,the existence of $L^p$-Lebesgue points quasieverywhere with respect to theBesov capacity, embeddings into H\"older spaces for appropriate exponents, anda stronger Lebesgue point result under an additional reverse doublinghypothesis on the measure. We also obtain several Poincar\'e-type inequalitiesrelating integrals of Besov functions over balls to integrals of uppergradients of extension of these functions to a uniformized hyperbolic fillingof the space.

Extension and trace theorems for noncompact doubling spaces