Extension and trace theorems for noncompact doubling spaces

Abstract

We generalize the extension and trace results of Björn-Björn-Shanmugalingam [3] 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 [7]. We deduce several new corollaries for the Besov spaces that arise as trace spaces in this fashion, including density of Lipschitz functions with compact support, embeddings into Hölder spaces for appropriate exponents, and the existence of Lebesgue points quasieverywhere with respect to the Besov capacity under an additional reverse doubling hypothesis on the measure. Under this same reverse doubling hypothesis we also obtain a hyperbolic Poincaré inequality for Besov functions that controls the mean oscillation of a Besov function over a ball through an integral over an extended ball of an upper gradient for an extension of this function to a uniformized hyperbolic filling of the space.

Characterizations for the existence of traces of first-order Sobolev spaces on hyperbolic fillings

“…By [3,Proposition 4.4], we know that the uniformized boundary ∂X and Z are biLipschitz equivalent, and thus, in the following, we identify ∂X and Z via the biLipschitz mapping. On X, we also use the measure µ defined in (6) below, as a result, we get a metric measure space (X, d, µ).…”

“…In [3], Björn-Björn-Shanmugalingam proved that for each 1 ≤ p < ∞, the trace space of a Newtonian-Sobolev space N 1,p (X) is the Besov space B θ p,p (Z) (0 < θ < 1) when the underlying space Z is compact and doubling, and when the hyperbolic filling X is equipped with a measure defined as in ( 6) by taking ρ(t) = e −ǫp(1−θ)t ; see [3,Theorem 1.1] for details. Later, Butler [6] extended the extension and trace results in [3] by weakening the hypothesis "compactness" on Z to the one "completeness". Notice that the trace functions f defined in [3] are different from the ones T f in this paper (see [3] or Section 4 for the definition of f ).…”

Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces

“…In the process of proving Theorem 1.2 we formulate a useful criterion (Proposition 3.3) for checking that µ β,b is doubling on Xε,b . This criterion will be used to verify Theorem 1.4 below as well as some key claims in our followup work [7].…”

“…Our results allow us to construct a number of interesting new unbounded metric measure spaces supporting Poincaré inequalities. A particularly important example is uniformizations of hyperbolic fillings of unbounded metric spaces, which play a key role in our followup work [7] concerning extension and trace theorems for Besov spaces on noncompact doubling metric measure spaces.…”