Boundary modulus of continuity and quasiconformal mappings (Q2908734)

In [the reviewer and \textit{R. Näkki}, J. Lond. Math. Soc., II. Ser. 44, No. 2, 339--350 (1991; Zbl 0755.30026)] it was shown that if in a bounded domain \(D\) a quasiconformal mapping \(f:D \rightarrow \mathbb{R}^n\), continuous in \(\overline{D}\), satisfies NEWLINE\[NEWLINE|f(x) - f(y)| \leq M|x-y|^{\alpha}\quad\text{for all}\quad x,y \in \partial D,NEWLINE\]NEWLINE then NEWLINE\[NEWLINE|f(x) - f(y)| \leq M^*|x-y|^{\beta}\quad\text{for all}\quad x,y \in \overline{D},NEWLINE\]NEWLINE where \(\beta = \min (\alpha, K_I(f)^{1/(1-n)})\) and \(M^*\) depends only on \(M\), \(\alpha\), \(n\), \(K(f)\) and \(\mathrm{diam}(D)\). The authors extend this result for more general moduli \(\omega\) of continuity. Assuming that a non-negative, non-decreasing function \(\omega\) satisfies \(\omega(2t) \leq 2 \omega(t)\), \(t \geq 0\), the authors replace the boundary condition by \(|f(x) - f(y)| \leq \omega(|x-y|)\) for all \(x \in \partial D\) and \(y \in D\) and obtain a corresponding result. If, in addition, \(\omega\) satisfies \(\omega(t)/t^{\alpha} \leq M \max (1,\omega(s)s^{-\alpha})\), \(\alpha = K_I(f)^{1/(1-n)}\), for all \(0 < s < t < \mathrm{diam}(D)\), then \(|f(x) - f(y)| \leq \omega(|x-y|)\) for all \(x,y \in \partial D\) yields NEWLINE\[NEWLINE|f(x) - f(y)| \leq C \max (\omega(|x-y|), |x-y|^{\alpha})\quad\text{for all}\quad x,y \in \overline{D}.NEWLINE\]NEWLINE In the case that \(\partial D\) is uniformly perfect the authors extend a result of \textit{A. Hinkkanen} and the last author [Complex Variables, Theory Appl. 13, No. 3--4, 251--267 (1990; Zbl 0706.30017)] to this setting. The dependence of the constant \(C\) in the estimates is carefully analysed.