Extension Theory And Continuous Regularity For Quasiconformal Mappings

In this paper we study the extension theory of quasiconformal mappings andthe continuous regularity with respect to the euclidean or hyperbolic metricsof generalized quasiconformal mappings.Firstly, we study the hyperbolic distance between the BA extension andtriangle extension of a quasisymmetric homeomorphism. After building a newexplicit expression of the hyperbolic distance between two points of H, wecombine an estimate of normalized quasisymmetric homeomorphisms and aproperty of convex functions to obtain an asymptotically sharp estimate of thehyperbolic distance between the BA extension and triangle extension. Theresult improves the corresponding one given by Ibragimov.Secondly, we study the continuous regularity of (K, K ’)-quasiconformalmappings and (K, K ’)-quasiconformal harmonic mappings. We show that a(K, K ’)-quasiconformal mapping with an unbounded imagine is not necessaryto be H lder continuous, which is different from the case of bounded imagineobtained by Kalaj and Mateljevi. We also show that (K, K ’)-quasiconformalmappings of H onto itself is Lipschitz continuous with respect to a Euclid-ean metric or hyperbolic metric. Moreover, we obtain four equivalent condi-tions for a harmonic mapping of H onto itself to be (K, K ’)-quasiconformalmapping.Thirdly, we study the geometric characteristic of a multiply connecteddomain in the sense of a quasihyperbolic metric. We obtain a quasihyperbolicdistance formula of a point to the line and show that the quasihyperbolicgeodesic from a point to the line is perpendicular to the line. Using this resultwe answer the open question posed by Klén affirmatively in the case of twopunctured or three punctured plane domains and also generalize the estimategiven by Klén.