| 1. |
- Adamowicz, Tomasz, et al.
(författare)
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Prime ends for domains in metric spaces
- 2013
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Ingår i: Advances in Mathematics. - : Elsevier. - 0001-8708 .- 1090-2082. ; 238, s. 459-505
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.
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| 2. |
- Aikawa, Hiroaki, et al.
(författare)
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Dichotomy of global capacity density in metric measure spaces
- 2018
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Ingår i: Advances in Calculus of Variations. - : WALTER DE GRUYTER GMBH. - 1864-8258 .- 1864-8266. ; 11:4, s. 387-404
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Tidskriftsartikel (refereegranskat)abstract
- The variational capacity cap(p) in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E subset of R-n, infx is an element of R(n)cap(p)(E boolean AND B(x, r), B(x, 2r))/cap(p)(B(x, r), B(x, 2r)) is either zero or tends to 1 as r -amp;gt; infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincare inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R-n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.
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| 3. |
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| 4. |
- Björn, Anders, et al.
(författare)
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A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure
- 2006
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 134:3, s. 509-519
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Tidskriftsartikel (refereegranskat)abstract
- A. Baernstein II (Comparison of p-harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543-551, p. 548), posed the following question: If G is a union of m open arcs on the boundary of the unit disc D, then is w a,p(G)=w a,p(G), where w a,p denotes the p-harmonic measure? (Strictly speaking he stated this question for the case m=2.) For p=2 the positive answer to this question is well known. Recall that for p≠2 the p-harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense. The purpose of this note is to answer a more general version of Baernstein's question in the affirmative when 1G is the restriction to ∂D of a Sobolev function from W 1,p(C). For p≥2 it is no longer true that XG belongs to the trace class. Nevertheless, we are able to show equality for the case m=1 of one arc for all 1, using a very elementary argument. A similar argument is used to obtain a result for starshaped domains. Finally we show that in a certain sense the equality holds for almost all relatively open sets.
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| 5. |
- 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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| 6. |
- 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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| 7. |
- 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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| 8. |
- Björn, Anders, et al.
(författare)
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Geometric analysis on Cantor sets and trees
- 2017
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Ingår i: Journal für die Reine und Angewandte Mathematik. - : WALTER DE GRUYTER GMBH. - 0075-4102 .- 1435-5345. ; 725, s. 63-114
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Tidskriftsartikel (refereegranskat)abstract
- dUsing uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.
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| 9. |
- 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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| 10. |
- Björn, Anders, et al.
(författare)
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Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces
- 2008
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Ingår i: Houston Journal of Mathematics. - 0362-1588. ; 34:4, s. 1197-1211
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Tidskriftsartikel (refereegranskat)abstract
- We show that on complete doubling metric measure spaces X supporting a Poincare inequality, all Newton-Sobolev functions u are quasicontinuous, i.e. that for every epsilon > 0 there is an open set U subset of X such that C-p(U) < epsilon and the restriction of u to X\U is continuous. This implies that the capacity is an outer capacity.
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