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Quasisymmetry, measure and a question of Heinonen. (English) Zbl 0881.30020

Let \(F\) be a subset of the Euclidean space, \(\mathbb{R}^n\), \(n>1\) and \(g:F\to \mathbb{R}^n\) be a mapping. We say \(g\) is quasisymmetric if \(g\) is not constant and if there exists a homeomorphism \(\eta :[0,\infty )\to [0,\infty)\) such that \(|x-y|\leq t|x-z|\) implies \(|g(x) - g(y)|\leq \eta (t) |g(x) - g(z)|\) for all \(x, y, z\) in \(F\). For mappings defined on all of \(\mathbb{R}^n\) the quasisymmetry condition is equivalent to quasiconformality. It is well known that for \(n>1\) quasisymmetric maps are absolutely continuous with respect to Lebesgue measure, but that this is not true when \(n=1\) as was established in a classic example of Beurling and Ahlfors.
In this paper, Semmes proves that if \(F\) is a compact subset of Lebesgue measure zero in \(\mathbb{R}^n\), \(n>1\) and \(g : F\to \mathbb{R}^n\) is quasisymmetric then \(g(F)\) also has Lebesgue measure zero. His proof relies on a number of modifications and extensions of \(g\) to larger domains which progressively thicken \(F\). He proves results on metric doubling measures and finds means to relate these ideas to Gehring’s result on Jacobians of quasiconformal mappings. Semmes’ result is important for studying the boundary continuity of quasiconformal maps in space [see J. Heinonen, ibid. 12, No. 3, 783-789 (1996; reviewed below)].

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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