In this paper, we solve the
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Abstract 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.Free, publiclyaccessible full text available July 1, 2025 
We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space.This approach is applicable even if the space does not support any Poincar´e inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric mea sure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map.more » « lessFree, publiclyaccessible full text available May 12, 2025

Abstract Given a compact doubling metric measure space
X that supports a 2Poincaré inequality, we construct a Dirichlet form on that is comparable to the upper gradient energy form on$$N^{1,2}(X)$$ ${N}^{1,2}\left(X\right)$ . Our approach is based on the approximation of$$N^{1,2}(X)$$ ${N}^{1,2}\left(X\right)$X by a family of graphs that is doubling and supports a 2Poincaré inequality (see [20]). We construct a bilinear form on using the Dirichlet form on the graph. We show that the$$N^{1,2}(X)$$ ${N}^{1,2}\left(X\right)$ limit$$\Gamma $$ $\Gamma $ of this family of bilinear forms (by taking a subsequence) exists and that$$\mathcal {E}$$ $E$ is a Dirichlet form on$$\mathcal {E}$$ $E$X . Properties of are established. Moreover, we prove that$$\mathcal {E}$$ $E$ has the property of matching boundary values on a domain$$\mathcal {E}$$ $E$ . This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form$$\Omega \subseteq X$$ $\Omega \subseteq X$ ) on a domain in$$\mathcal {E}$$ $E$X with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects. 
Free, publiclyaccessible full text available December 1, 2024

Xiao, Jie ; Lenhart, Suzanne (Ed.)In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromovhyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space.more » « less

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

Abstract The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably pathconnected, noncomplete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally biLipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.more » « less

Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional pLaplacians (−∆p)θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non homogenous nonlocal equation (−∆p)θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for pLaplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality.more » « less