Global integrability of the Jacobian and quasiconformal maps (Q1330151)

Let \(f: D\to D'\) be a \(K\)-quasiconformal map, where \(D\) and \(D'\) are domains in \(\mathbb{R}^ n\). In 1973 \textit{F. W. Gehring} [Acta Math. 130, 265-277 (1973; Zbl 0258.30021)] proved that for some \(\varepsilon> 0\), \(J^{1+ \varepsilon}_ f\), where \(J_ f\) if the Jacobian of \(f\), is locally integrable in \(D\). More recently, \textit{K. Astala} and \textit{P. Koskela} [J. Anal. Math. 57, 203-220 (1991; Zbl 0772.30022)] answered the question, ``When is \(J^{1+ \varepsilon}_ f\) integrable over \(D\)?''. A crucial step in their argument involves establishing the equivalence of this question with the local Lipschitz question studied by \textit{F. W. Gehring} and \textit{O. Martio} [Ann. Acad. Sci. Fenn., Ser. A I 10, 203-219 (1985; Zbl 0584.30018)]. In the paper under review, the author presents direct proofs of the global integrability of the Jacobian results. These proceed using quasihyperbolic metric and BMO estimates. Furthermore, these proofs lead to ideas and techniques useful in the study of the domains involved.