On Kellogg's theorem for quasiconformal mappings (Q2902680)

A homeomorphism \(f:\Omega\to\Omega'\) between planar domains is called \(K\)-quasiconformal if it preserves orientation, belongs to the Sobolev class \(W_{\text{loc}}^{1,2}(\Omega)\), and its directional derivatives satisfies the distortion inequality \(\max_\alpha|\partial_\alpha f|\leq K\min_\alpha|\partial_\alpha f|\) a.e. in \(\Omega\). This estimate is equivalent to saying that \(f\) satisfies the Beltrami equation \(\bar{\partial}f(z)=\mu(z)\partial f(z)\) for almost all \(z\in\mathbb R^2\), where \(\mu\) is the so-called Beltrami coefficient or dilatation with \(\|\mu\|_\infty\leq k=(K-1)/(K+1)\).NEWLINENEWLINEIn the paper under review, the author generalizes the classical results for a class of quasiconformal mappings due to Kellogg and Warschawski. The author proves that a quasiconformal mapping \(f\) between two planar domains with smooth \(C^{1,\alpha}\) (\(0<\alpha<1\)) boundaries together with its inverse mapping \(f^{-1}\) is \(C^{1,\alpha}\) up to the boundary if and only if the Beltrami coefficient \(\mu_f\) is uniformly \(\alpha\) Hölder continuous.