Search results for "30c65"

showing 4 items of 34 documents

Mappings of finite distortion : boundary extensions in uniform domains

2015

In this paper, we consider mappings on uniform domains with exponentially integrable distortion whose Jacobian determinants are integrable. We show that such mappings can be extended to the boundary and moreover these extensions are exponentially integrable with quantitative bounds. This extends previous results of Chang and Marshall on analytic functions, Poggi-Corradini and Rajala and Akkinen and Rajala on mappings of bounded and finite distortion.

mappings of finite distortionJohn domainMathematics - Complex Variablesradial limitsFOS: Mathematicsquasiregular mappingsweighted capacityComplex Variables (math.CV)uniform domain30C65
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Uniformization of two-dimensional metric surfaces

2014

We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the Euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by qu…

metric surfacesPure mathematicsMathematics - Complex VariablesGeneral Mathematics010102 general mathematicsPrimary 30L10 Secondary 30C65 28A75 51F99 52A38Hausdorff spaceMetric Geometry (math.MG)01 natural sciencesUpper and lower boundsMetric spaceMathematics - Metric GeometryBounded function0103 physical sciencesMetric (mathematics)Euclidean geometryFOS: MathematicsMathematics::Metric Geometry010307 mathematical physicsComplex Variables (math.CV)0101 mathematicsUniformization (set theory)ParametrizationMathematicsInventiones mathematicae
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Quasiconformal Jordan Domains

2020

We extend the classical Carath\'eodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has finite Hausdorff $2$-measure, the boundary $\partial Y = \overline{Y} \setminus Y$ is homeomorphic to $\mathbb{S}^{1}$, and there exists a homeomorphism $\phi \colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $\phi$ has a continuous, monotone, and surjective extension $\Phi \colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. This result is best possible in this generality. In addition, we find a n…

primary 30l10QA299.6-433Mathematics::Dynamical SystemsMathematics - Complex VariablesMathematics::Complex VariablesHigh Energy Physics::PhenomenologycarathéodoryPrimary 30L10 Secondary 30C65 28A75 51F99 52A38Mathematics::General Topologymetric surfacebeurling–ahlforsMetric Geometry (math.MG)quasiconformalsecondary 30c65 28a75 51f99Carathéodorymetriset avaruudetfunktioteoriaPhysics::Fluid DynamicsMathematics - Metric GeometryBeurling–AhlforsFOS: MathematicsmittateoriaComplex Variables (math.CV)AnalysisAnalysis and Geometry in Metric Spaces
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Weighted Hardy Spaces of Quasiconformal Mappings

2019

We establish a weighted version of the $H^p$-theory of quasiconformal mappings.

radial maximal functionsfunktioteoriaHardy spacesMathematics - Complex Variablesmodulus estimateHardyn avaruudetFOS: Mathematicsquasiconformal mappingGeometry and TopologyComplex Variables (math.CV)nontangential30C65The Journal of Geometric Analysis
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