| 1. |
- Björn, Anders, et al.
(författare)
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Bounded Geometry andp-Harmonic Functions Under Uniformization and Hyperbolization
- 2021
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Ingår i: Journal of Geometric Analysis. - : SPRINGER. - 1050-6926 .- 1559-002X. ; 31, s. 5259-5308
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Tidskriftsartikel (refereegranskat)abstract
- The uniformization and hyperbolization transformations formulated by Bonk et al. in"Uniformizing Gromov Hyperbolic Spaces", Asterisque, 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 localp-Poincare inequality, then the transformed measure is globally doubling and supports a globalp-Poincare 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 localp-Poincare inequality, carries nonconstant globally definedp-harmonic functions with finitep-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.
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| 2. |
- Björn, Anders, 1966-, et al.
(författare)
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Classification of metric measure spaces and their ends using p-harmonic functions
- 2022
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Ingår i: Annales Fennici Mathematici. - : SUOMALAINEN TIEDEAKATEMIA. - 2737-0690 .- 2737-114X. ; 47:2, s. 1025-1052
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Tidskriftsartikel (refereegranskat)abstract
- 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-Poincare 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 well-separated p-hyperbolic sequences of sets towards infinity. We also show that every such space X has a function f is an element of/ LP(X) + R with finite p-energy.
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| 3. |
- Björn, Anders, et al.
(författare)
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Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
- 2022
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Ingår i: Journal des Mathématiques Pures et Appliquées. - : Elsevier. - 0021-7824 .- 1776-3371. ; 159, s. 196-249
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study connections between Besov spaces of functions on a compactmetric space Z, equipped with a doubling measure, and the Newton–Sobolev spaceof functions on a uniform domain Xε. This uniform domain is obtained as auniformization of a (Gromov) hyperbolic filling of Z. To do so, we construct afamily of hyperbolic fillings in the style of Bonk–Kleiner [9] and Bourdon–Pajot [13]. Then for each parameter β > 0 we construct a lift μβ of the doubling measure νon Z to Xε, and show that μβ is doubling and supports a 1-Poincaré inequality.We then show that for each θ with 0 < θ < 1 and p ≥ 1 there is a choice of β = p(1 − θ)ε such that the Besov space is the trace space of the Newton–Sobolev space N1,p(Xε, μβ). Finally, we exploit the tools of potential theory on Xεto obtain fine properties of functions in , such as their quasicontinuity andquasieverywhere existence of Lq-Lebesgue points with q = sνp/(sν − pθ), where sν is a doubling dimension associated with the measure ν on Z. Applying this tocompact subsets of Euclidean spaces improves upon a result of Netrusov [43] in .
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| 4. |
- Björn, Anders, et al.
(författare)
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Locally p-admissible measures on R
- 2020
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Ingår i: Journal of Functional Analysis. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0022-1236 .- 1096-0783. ; 278:4
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Tidskriftsartikel (refereegranskat)abstract
- In this note we show that locally p-admissible measures on R necessarily come from local Muckenhoupt A(p) weights. In the proof we employ the corresponding characterization of global p-admissible measures on R in terms of global A(p) weights due to Bjorn, Buckley and Keith, together with tools from analysis in metric spaces, more specifically preservation of the doubling condition and Poincare inequalities under flattening, due to Durand-Cartagena and Li. As a consequence, the class of locally p-admissible weights on R is invariant under addition and satisfies the lattice property. We also show that measures that are p-admissible on an interval can be partially extended by periodical reflections to global p-admissible measures. Surprisingly, the p-admissibility has to hold on a larger interval than the reflected one, and an example shows that this is necessary. (C) 2019 Elsevier Inc. All rights reserved.
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| 5. |
- Björn, Anders, et al.
(författare)
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The Liouville theorem for p-harmonic functions and quasiminimizers with finite energy
- 2021
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Ingår i: Mathematische Zeitschrift. - : SPRINGER HEIDELBERG. - 0025-5874 .- 1432-1823. ; 297:1-2, s. 827-854
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Tidskriftsartikel (refereegranskat)abstract
- We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite pth power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global p-Poincare 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 R, we characterize all locally doubling measures, supporting a local p-Poincare inequality, for which there exist nonconstant quasiminimizers of finite p-energy, and show that a quasiminimizer is of finite p-energy if and only if it is bounded. As p-harmonic functions are quasiminimizers they are covered by these results.
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