Asymptotic values and Hölder continuity of quasiconformal mappings
A variety of Hölder estimates for quasiconformal maps of the unit ball \(B^ n\) into \(R^ n\) on sets A are obtained from similar estimates on essentially thiner sets than A. As asymptotic value estimate: Let E be an arc in \(B^ n\) with an endpoint \(b\in \partial B^ n\). Suppose that for some \(b'\in R^ n\), \[ (*)\quad | f(x)-b'| \leq M| x- b|^{\alpha}, \] for all \(x\in E\), \(\alpha >0\). Then for each angle \(\theta\) in (0,\(\pi\) /2) there is a constant \(M_{\theta}\) such that \[ | f(x)-b'| \leq M_{\theta}| x-b|^{\alpha} \] whenever x lies in the Stolz cone \(S_{\theta}(b)\). This result can be used to prove: Let \(b_ 1,...,b_ p\) be distinct boundary points of \(B^ n\) and let (*) hold for each \(b_ i\) with \(\alpha =\alpha_ i\). Then \[ \alpha_ 1^{n-1}+...+\alpha_ p^{n-1}\leq 2K_ 0(f). \] A Hardy- Littlewood type theorem: Let \(A_ f\) be the set of points where the radial limit of f exists and write \(\tilde f\) for the associated boundary function. If \(\tilde f\in Lip_{\alpha}(A_ f)\), then \(f\in Lip_{\beta}(B^ n)\) for \(\beta =\min (\alpha,K_ I(f)^{1/(1-n)})\). Another approach to this latter type of problems is given in \textit{[F. W. Gehring and \textit{O. Martio}}, Ann. Acad. Sci. Fenn., Ser. A I 10, 203-219 (1985; Zbl 0584.30018)].
Hölder continuity of quasiconformal mappings
scientific article; zbMATH DE number 4029903
scientific article