Definitions of quasiconformality
Let \(f : D \to \mathbb{R}^n\), \(D \subset \mathbb{R}^n\) a domain, be a homeomorphism. The mapping \(f\) is quasiconformal iff \((*)\) \(\limsup_{n \to 0} H(x,f,r) \leq C < \infty\) for all \(x \in D\); here \(H(x,f,r) = L(x,f,r)/l(x,f,r)\) and \(L(x,f,r) = \max \{|f(y) - f(x) |: |y - x |= r\}\); for \(l(x,f,r) \max\) is replaced by min. No simple proof for \(\Leftarrow\) is known. The \(H\)-dilatation can be used to study homeomorphisms between arbitrary metric spaces \(X\) and \(Y\). The authors look for global estimates which can be derived from the boundedness of \(H\). Such an estimate is the quasisymmetry condition [\textit{P. Tukia} \textit{J. Väisälä}: Ann. Acad. Sci. Fenn., Ser. A I 5, 97-114 (1980; Zbl 0443.54011)], which holds in the case \(X = \mathbb{R}^n = Y\). The authors show that this conclusion remains true if \(X = Y\) is a Garnot group [A. Korányi, H. M. Riemann: Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. (to appear)]. As a byproduct they show that the lim sup in \((*)\) can be replaced by the lim inf. The proofs employ elementary covering arguments instead of the usual capacity estimates.
scientific article; zbMATH DE number 746069
scientific article