Analytic properties of locally quasisymmetric mappings from Euclidean domains (Q2710408)

Let \(G\) be a domain in \(\mathbb{R}^n\) and \(Y\) a metric space. A homeomorphism \(f:G\to Y\) is called locally quasisymmetric if each point in \(G\) has a neighborhood \(U\) where \(f\) is \(\eta\)-quasisymmetric, i.e. \(|a-x|\leq t|b-x|\) implies \(|f(a)-f(x) |\leq\eta (t)|f(b)-f(x) |\) whenever \(t>0\), \(a,b,x\in U\) and \(\eta:[0,\infty) \to[0,\infty)\) is a homeomorphism. Assuming that \(Y\) has locally finite Hausdorff \(n\)-measure the author shows that \(f\) is \(ACL\) and that \(f\) is a Sobolev function in the sense of [\textit{Yu. G. Reshetnyak}, Sib. Mat. Zh. 38, No. 3, 657-675 (1997; Zbl 0944.46024)]. \(f\) also satisfies the Lusin condition \((N)\) with respect to the fine \(n\)-packing measure in \(Y\) and thus with respect to the \(n\)-Hausdorff measure in \(Y\) as well. These results are extensions of the results of [\textit{J. Väisälä}, Trans. Am. Math. Soc. 264, 191-204 (1981; Zbl 0456.30018)] where it was assumed that \(Y\) was embedded in some (high) dimensional Euclidean space. Most of these results are not known if \(G\) is replaced by a metric space \(X\) although so features can be developed when \(X\) is an Ahlfors regular space with some extra assumptions.