Search results for "conformal mapping"

Analytic Properties of Quasiconformal Mappings Between Metric Spaces

2012

We survey recent developments in the theory of quasiconformal mappings between metric spaces. We examine the various weak definitions of quasiconformality, and give conditions under which they are all equal and imply the strong classical properties of quasiconformal mappings in Euclidean spaces. We also discuss function spaces preserved by quasiconformal mappings.

Global Lp -integrability of the derivative of a quasiconformal mapping

1988

Let f be a quasiconformal mapping of an open bounded set U in Rn into Rn . Then f′ belongs to Lp(U) for some p > n provided that f satisfies (a) U is a uniform domain and fU is a John domain or (b) f is quasisymmetric and U satisfies a metric plumpness condition.

Hölder continuity of Sobolev functions and quasiconformal mappings

1993

Quasiconformal distortion on arcs

1994

Distortion of quasiconformal maps in terms of the quasihyperbolic metric

2013

Abstract We extend a theorem of Gehring and Osgood from 1979–relating to the distortion of the quasihyperbolic metric by a quasiconformal mapping between Euclidean domains–to the setting of metric measure spaces of Q -bounded geometry. When the underlying target space is bounded, we require that the boundary of the image has at least two points. We show that even in the manifold setting, this additional assumption is necessary.

Geometric Properties of Planar BV -Extension Domains

2009

We investigate geometric properties of those planar domains that are extension for functions with bounded variation.We start from a characterization of such domains given by Burago–Maz'ya and prove that a bounded, simply connected domain is a BV -extension domain if and only if its com- plement is quasiconvex. We further prove that the extension property is a bi-Lipschitz invariant and give applications to Sobolev extension domains.

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

2011

Abstract In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces B ˙ p , q s and Triebel–Lizorkin spaces F ˙ p , q s for all s ∈ ( 0 , 1 ) and p , q ∈ ( n / ( n + s ) , ∞ ] , both in R n and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve F ˙ n / s , q s on R n for all s ∈ ( 0 , 1 ) and q ∈ ( n / ( n + s ) , ∞ ] . A metric measure space version of the above morphism property is also established.

On a fundamental variational lemma for extremal quasiconformal mappings

1986

Lectures on quasiconformal and quasisymmetric mappings

2009

A quasiconformal composition problem for the Q-spaces

2017

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_{\alpha}(\mathbb R^n)$ depends on the index $\alpha$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $\alpha$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-V\"ais\"al\"a's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitr…