Studies On The Characteristics Of Quasihyperbolic Mappings In Banach Spaces

In 1976,Gehring and Palka introduced the concept of the quasihyperbolic metric,which has been widely concerned and applied now,it became an important research tool of the quasiconformal mappings.In the 1990s,V(?)is(?)l(?) established the theory of the free quasiconformal mappings in infinite dimensional Banach spaces by using quasihyperbolic metric.In this paper,we mainly consider the properties of the quasihyperbolic mappings(Bilipschitz mappings under the quasihyperbolic metric)in Banach spaces.The full paper consists of three chapters,as follows:In the first chapter,we introduce the background of our problems,some related concepts and properties,and our main results in this paper.In the second chapter,we first provide an example to show that the local quasisymmetric mappings are not the necessary conditions for the quasihyperbolic mappings.Meanwhile,we give three equivalent characterizations of the quasihyperbolic mappings in Banach spaces,and give the proofs of them.In the third chapter,we study the subinvariance property of John domains under the quasihyperbolic mappings in Banach spaces,and we obtain that the image of any John subdomain in the preimage domain is also a John domain under the following conditions,that is,the image domain is a uniform domain,and the mappings are quasihyperbolic and quasisymmetric relative to the boundary of domain.