Recent zbMATH articles in MSC 30C65
https://www.zbmath.org/atom/cc/30C65
2022-05-16T20:40:13.078697Z
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Weighted Hardy spaces of quasiconformal mappings
https://www.zbmath.org/1483.30055
2022-05-16T20:40:13.078697Z
"Benedict, Sita"
https://www.zbmath.org/authors/?q=ai:benedict.sita
"Koskela, Pekka"
https://www.zbmath.org/authors/?q=ai:koskela.pekka
"Li, Xining"
https://www.zbmath.org/authors/?q=ai:li.xining
Summary: We study the integral characterizations of weighted Hardy spaces of quasiconformal mappings on the \(n\)-dimensional unit ball using the weight \((1-r)^{n-2 + \alpha}\). We extend the known results for univalent functions on the unit disk. Some of our results are new even in the unweighted setting for quasiconformal mappings.
Sphericalization and flattening preserve uniform domains in nonlocally compact metric spaces
https://www.zbmath.org/1483.30056
2022-05-16T20:40:13.078697Z
"Li, Yaxiang"
https://www.zbmath.org/authors/?q=ai:li.yaxiang
"Ponnusamy, Saminathan"
https://www.zbmath.org/authors/?q=ai:ponnusamy.saminathan
"Zhou, Qingshan"
https://www.zbmath.org/authors/?q=ai:zhou.qingshan
Summary: The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-MÃ¶bius maps preserve uniform domains in nonlocally compact spaces as well.
Weak quasicircles have Lipschitz dimension 1
https://www.zbmath.org/1483.30101
2022-05-16T20:40:13.078697Z
"Freeman, David M."
https://www.zbmath.org/authors/?q=ai:freeman.david-mandell
Summary: We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, the Lipschitz dimension of any weak quasicircle or arc is equal to 1.
Quasiconformal Jordan domains
https://www.zbmath.org/1483.30102
2022-05-16T20:40:13.078697Z
"Ikonen, Toni"
https://www.zbmath.org/authors/?q=ai:ikonen.toni
Summary: We extend the classical CarathÃ©odory extension theorem to quasiconformal Jordan domains \((Y, d_Y)\). We say that a metric space \((Y, d_Y)\) is a \textit{quasiconformal Jordan domain} if the completion \(\overline{Y}\) of \((Y, d_Y)\) has finite Hausdorff 2-measure, the \textit{boundary} \(\partial Y = \overline{Y}\setminus Y\) is homeomorphic to \(\mathbb{S}^1\), and there exists a homeomorphism \(\phi : \mathbb{D} \rightarrow (Y, d_Y)\) that is quasiconformal in the geometric sense.
We show that \(\phi\) has a continuous, monotone, and surjective extension \(\Phi : \overline{\mathbb{D}} \rightarrow \overline{Y}\). This result is best possible in this generality. In addition, we find a necessary and sufficient condition for \(\Phi\) to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of \(\Phi\) to \(\mathbb{S}^1\) being a quasisymmetry and to \(\partial Y\) being bi-Lipschitz equivalent to a quasicircle in the plane.
Uniformization of metric surfaces using isothermal coordinates
https://www.zbmath.org/1483.30103
2022-05-16T20:40:13.078697Z
"Ikonen, Toni"
https://www.zbmath.org/authors/?q=ai:ikonen.toni
Summary: We establish a uniformization result for metric surfaces -- metric spaces that are topological surfaces with locally finite Hausdorff \(2\)-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.
Quasisymmetric Koebe uniformization with weak metric doubling measures
https://www.zbmath.org/1483.30105
2022-05-16T20:40:13.078697Z
"Rajala, Kai"
https://www.zbmath.org/authors/?q=ai:rajala.kai
"Rasimus, Martti"
https://www.zbmath.org/authors/?q=ai:rasimus.martti
Summary: We give a characterization of metric spaces quasisymmetrically equivalent to a finitely connected circle domain. This result generalizes the uniformization of Ahlfors 2-regular spaces by \textit{S. Merenkov} and \textit{K. Wildrick} [Rev. Mat. Iberoam. 29, No. 3, 859--910 (2013; Zbl 1294.30043)].