On mappings with constraints on the measure of dilatation (Q1288125)

Let \(f:\mathbb C\to \mathbb C\) be an orientation preserving quasiconformal homeomorphism, \(f\in W^1_{1,\text{loc}}\) with constraints on the measure of dilatation in the form \(\text{mes}\{z\in \mathbb C:p(z)>t \} \leq \varphi(t)\), where \(p(z)=(1+| \mu(z)|)/(1-| \mu(z)|)\), \(\mu\:{\mathbb C}\to{\mathbb C}\) is a measurable function, \(\varphi\) is an arbitrary function. Necessary and sufficient conditions for compactness of such classes are established. For instance, it is shown that almost all such classes are noncompact. In particular, a negative answer is given concerning compactness of David classes [\textit{G. David}, Ann. Acad. Sci. Fenn., Ser. AI 13, 25-70 (1988; Zbl 0619.30024)].