Quasiconformal mappings and global integrability of the derivative
If \(f: D\to D'\) is a bounded \(K\)-quasiconformal mapping, then \(\int_ D | f'(x)|^ n dx\leq K| D'|<\infty\). The authors study conditions which imply a global \(L^ p\)-integrability of the derivative for \(p>0\). For example, it is shown that \(f'\) is in \(L^ p\), \(p>n\), if \(D\) is a uniform domain and \(D'\) satisfies a quasihyperbolic boundary condition. This extends a result of [the reviewer and \textit{J. Väisälä}, Complex Variables, Theory Appl. 9, No. 4, 309-319 (1988; Zbl 0629.30023)]. For conformal mappings \(f: B(0,1)\to\mathbb{C}\) this result leads to an equivalence: \(f'\in L^ p(B(0,1))\), \(p>2\), iff \(| f(z)- f(w)|\leq M| z-w|^ \alpha\) for some \(\alpha>0\) and \(M<\infty\). The paper also provides a good account of integrability results for \(p<n\).
scientific article; zbMATH DE number 221309
scientific article