Some properties of quasisymmetries in metric spaces

Abstract

Let \((X, d, \mu)\) and \((Y, d', \mu')\) be metric spaces \(\alpha\)-regular by Ahlfors with \(\alpha>0\) and locally finite Borel measures \(\mu\) and \(\mu'\), respectively. We consider the class \(ACS_{E}\) of absolutely continuous functions on a.a. compact subsets \(E\subset X\) and establish the membership of mappings \(f: X\to Y\) to a given class.