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By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite \(p\)energy \(p\)harmonic and \(p\)quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local \(p\)Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the \(p\)hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant \(p\)harmonic functions with finite \(p\)energy as spaces having at least two wellseparated \(p\)hyperbolic sequences of sets towards infinity. We also show that every such space \(X\) has a function \(f \notin L^p(X) + \mathbf{R}\) with finite \(p\)energy.more » « less

null (Ed.)In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a 1Poincare inequality. We show that at almost every point x outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at x. We also show that, at codimension 1 Hausdorff measure almost every measuretheoretic boundary point of a set E of finite perimeter, there is an asymptotic limit set (E)∞ corresponding to the asymptotic expansion of E and that every such asymptotic limit (E)∞ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of (E)∞ is Ahlfors codimension 1 regular.more » « less

null (Ed.)In this note we show that given two complete geodesic Gromov hyperbolic spaces that are roughly isometric and an arbitrary (not necessarily small), either the uniformization of both spaces with parameter epsilon results in uniform domains, or else neither uniformized space is a uniform domain. The terminology of "uniformization" is from [BHK], where it is shown that the uniformization, with parameter epsilon, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided epsilon is small enough.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

null (Ed.)Abstract We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite p th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $$p$$ p Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $$\mathbf {R}$$ R , we characterize all locally doubling measures, supporting a local $$p$$ p Poincaré inequality, for which there exist nonconstant quasiminimizers of finite $$p$$ p energy, and show that a quasiminimizer is of finite $$p$$ p energy if and only if it is bounded. As $$p$$ p harmonic functions are quasiminimizers they are covered by these results.more » « less