Radial length, radial John disks and $K$-quasiconformal harmonic mappings

On certain quasiconformal and elliptic mappings

Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings $f$ satisfying the Poisson's equation $Δf=g$ in $\mathbb{D}$, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.

Problems in holomorphic dynamics

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichmüller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of Julia sets; John Milnor, Problems on local connectivity. 3. Measurable Dynamics: Mikhail Lyubich, Measure and Dimension of Julia Sets; Feliks Przytycki, On invariant measures for iterations of holomorphic maps. 4. Iterates of Entire Functions: Robert De…

Loewner chains with quasiconformal extensions: an approximation approach

A new approach in Loewner Theory proposed by Bracci, Contreras, Díaz-Madrigal and Gumenyuk provides a unified treatment of the radial and the chordal versions of the Loewner equations. In this framework, a generalized Loewner chain satisfies the differential equation $$ \partial_{t}f_{t}(z) = (z - τ(t))(1-\overline{τ(t)}z)\partial_{z}f_{t}(z)p(z,t), $$ where $τ: [0,\infty) \to \overline{\mathbb{D}}$ is measurable and $p$ is called a Herglotz function. In this paper, we will show that if there exists a $k \in [0,1)$ such that $p$ satisfies $$ |p(z,t) - 1| \leq k |p(z,t) + 1| $$ for all $z \in \…

Hardy spaces and quasiconformal maps in the Heisenberg group

We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Korányi unit ball $B$ in the first Heisenberg group $\mathbb{H}^1$. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of $p_0(K)>0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential …

关于调和拟共形环带映射的 Iwaniec--Kovalev--Onninen 猜想

On an Iwaniec--Kovalev--Onninen Conjecture for harmonic quasiconformal annulus mappings

本文构造非径向调和拟共形环带映射，证明 Iwaniec--Kovalev--Onninen 猜想中的伸缩商下界不成立，并给出单模估计成立的若干条件。

Norm inequalities for vector functions

We study vector functions of ${\mathbb R}^n$ into itself, which are of the form $x \mapsto g(|x|)x\,,$ where $g : (0,\infty) \to (0,\infty) $ is a continuous function and call these radial functions. In the case when $g(t) = t^c$ for some $c \in {\mathbb R}\,,$ we find upper bounds for the distance of image points under such a radial function. Some of our results refine recent results of L. Maligranda and S. Dragomir. In particular, we study quasiconformal mappings of this simple type and obtain norm inequalities for such mappings.