 Award ID(s):
 2141297
 NSFPAR ID:
 10519882
 Publisher / Repository:
 Arxiv
 Date Published:
 Format(s):
 Medium: X
 Institution:
 Arxiv
 Sponsoring Org:
 National Science Foundation
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Abstract We consider two notions of functions of bounded variation in complete metric measure spaces,one due to Martio and the other due to Miranda Jr. We show that these two notionscoincide if the measure is doubling and supports a 1Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1Poincaré inequality, then the metric space supports a Semmes family of curves structure.more » « less

null (Ed.)Abstract The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces” , Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p Poincaré inequality, then the transformed measure is globally doubling and supports a global p Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p Poincaré inequality, carries nonconstant globally defined p harmonic functions with finite p energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain.more » « less

In this paper, we generalize a biLipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces X so that any biLipschitz embedding of a subset of the real line into X extends to a biLipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset Y of X has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in X by biLipschitz curves.more » « less

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if
$(X,d,\mu )$ is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space$N^{1,p}(X)$ . Here the measure$\mu$ is Borel and is finite and positive on all metric balls. In particular, we don’t assume properness of$X$ , doubling of$\mu$ or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply tolocally complete spaces$X$ and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity. 
Abstract In this paper, we solve the
p Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameterp . We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adamstype inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.