Studies On The Relationship Between Quasihyperbolic Mappings And Bilipschitz Mappings In Banach Spaces

It is well known that quasiconformal mappings are generalizations of conformal mappings.This concept was first proposed by Gr¨otzsch.After the introduction of this concept,it attracts much attention and has become an important research topic.In the 1990 s,V¨ais¨al¨a established the theory of quasiconformal mappings in Banach spaces,and in this way,quasiconformal mappings have been generalized into the case of Banach spaces with infinity dimension from Euclidean spaces.Quasihyperbolic mappings are quasiconformal mappings.The purpose of this thesis is to discuss the extension property of quasihyperbolic mappings in Banach spaces.This thesis consists of four chapters,and its arrangement is as follows.In Chapter one,we present the background of our research problems and state our main result.In Chapter two,we introduce the necessary terminology and known related results.In Chapter three,we give several lemmas and corollaries needed in the proof of the main results.In Chapter four,we discuss the extension property of Bilipschitz mappings in Banach spaces,and prove that for quasihyperbolic mapping on a bounded domain in Banach spaces,if it is Bilipschitz on the boundary,then it is Bilipschitz in the whole domain with certain additional conditions.