Nonclosed David classes (Q1594129)

This paper deals with orientation preserving homeomorphisms \(f\) of the plane that belong to the Sobolev class \(W^1_{1,loc}\), i.e., solutions of the Beltrami equation \(f_{\overline z}=\mu(z)f_z\), where \(\mu:\mathbb{C}\to \mathbb{C}\) is a measurable function with \(|\mu(z) |\leq 1\) and (1) \(f(0)=0\), \(f(1)=1\), \(f (\infty) =\infty\). The dilatation of the mapping is \(p(z)=(1+ |\mu(z) |) /(1-|\mu(z)|)\). The authors consider the following general measure constraints on the dilatation (2) \(\text{mes}\{z\in\mathbb{C}:p(z)> t\}\leq \varphi (t)\), where \(\varphi:I \to\overline R^+\), \(I=[1,\infty)\) is an arbitrary function. The main result is the following theorem: Theorem 1. Suppose that a function \(\varphi :I \to\overline R^+\) does not increase and is right-continuous. Then the class of all orientation preserving homeomorphisms of the plane normalized by (1) and satisfying (2) is sequentially compact if and only if \(\varphi\) has the form \(\varphi(t)= \begin{cases}\infty,\quad & 1\leq t<Q\\ 0,\quad & t\geq Q\end{cases}\) for some \(1\leq Q<\infty\). The proof is not outlined.