BMO-quasiconformal mappings (original) (raw)

2001, Journal d'Analyse Mathématique

Plane BMO-quasiconforrnal and BMO-quasiregular mappings are introduced, and their basic properties are studied. This includes distortion, existence, uniqueness, representation, integrability, convergence and removability theorems, the reflection principle, boundary behavior and mapping properties.

Criteria of convergence for quasiconformal mappings and their generalizations

Ukrainian Mathematical Journal, 1996

We establish necessary and sufficient conditions for the convergence of normalized homeomorphisms of Sobolev class in terms of the Fourier transforms of complex characteristics in the case where the upper bound of dilations is exponentially bounded in measure. This allows us to construct various metrics generating locally uniform convergence of mappings.

On the area distortion by quasiconformal mappings

Proceedings of the American Mathematical Society, 1995

We give the sharp constants in the area distortion inequality for quasiconformal mappings in the plane. Astala [I] proved the following theorem conjectured by Gehring and Reich in [3]: Theorem A. Let f be a K-quasiconformal mapping of D = { z : lzl < 1) onto itself with f ( 0 )= 0 . Then for any measurable E c D we have where I I stands for the area. The first author [2] obtained a shorter proof which did not make use of the elaborate Thermodynamic Formalism and Holomorphic Motion Theory of the original proof of Astala. In late 1992 the second author [4] circulated a minimal proof which gives sharp bounds for the constants under the normalization f E C ( K ), i.e. f is a K-quasiconformal mapping of the plane which is conformal on C\o and f ( z )= z +o(1) near oo . In the interests of having a short sharp proof we combined our efforts.

Mapping problems for quasiregular mappings

2012

Abstract: We study images of the unit ball under certain special classes of quasiregular mappings. For homeomorphic, ie, quasiconformal mappings problems of this type have been studied extensively in the literature. In this paper we also consider non-homeomorphic quasiregular mappings. In particular, we study (topologically) closed quasiregular mappings originating from the work of J. V\" ais\" al\" a and M. Vuorinen in 1970's. Such mappings need not be one-to-one but they still share many properties of quasiconformal mappings.

Quasiconformal mappings and spaces of functions with generalized first …

Siberian Mathematical Journal

Let G be a fixed domain in R n. A condenser in G is a pair of connected compacta F0, F~ ~ G whose intersection is empty. A continuous function u: G ~ R having generalized first derivatives is said::to be ad- missible for the condenser if u(x) _< 0 on F0, u(x) -> 1 on F l, and tf~e ...

A Commentary on Teichm{\"u}ller's paper"Untersuchungen \"uber konforme und quasikonforme Abbildungen"(Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the \emph{Handbook of Teichm\"uller theory}

2019

This is a commentary on Teichmüller’s paper Untersuchungen über konforme und quasikonforme Abbildungen (Investigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry, in particular a lemma, known as the Modulsatz, which insures the almost circularity of certain loci defined as complementary components of simply connected regions in the Riemann sphere, and another lemma, which we call the Main Lemma, which insures the circularity near infinity of the image of circles by a quasiconformal map. The two results find wide applications in value distribution theory, where they allow the efficient use of moduli of doubly connected domains and of quasiconformal mappings. Teichmüller’s paper also contains a thorough development of the theory of conformal invariants of doubly connected domains. The final version of this paper will appear in Vol. VII of the Handbook of Teichmüller theory (European Mathematical Society Pub...

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