Complex Analysis and Operator Theory

, Volume 5, Issue 3, pp 967-984

First online:

Extendability of Classes of Maps and New Properties of Upper Sets

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We continue to study upper sets \({\widetilde{A}=\{(x,r)\in A\times R_+ :\exists y\in A\setminus\{x\}, r=|x-y|\}}\) equipped by hyperbolic metric. We define analogous of quasiconvexity, simply connectedness and nearlipschitz functions. We give a new definition of quasisymmetry as nearlipschitz characteristic on \({\widetilde{A}}\). In the final part in terms of upper sets we give the following extension property of \({A\subset R^2}\). For \({0\le\varepsilon\le \delta}\), each \({(1+\varepsilon)}\)-bilipschitz map f : AR 2 has an extension to a \({(1+C\varepsilon)}\)-bilipschitz map F : R 2R 2.