Studies On The Characterizations Of The Free Quasiconformal Mappings In Banach Spaces

The quasiconformal mapping was proposed by Grotzsch when he studied the Riemann mapping theorem from square to rectangle on the plane,which preserves the vertex correspondence.Then,this concept was extended to the euclidean space Rn,which is the generalization of conformal mapping.In the 1990s,V(?)is(?)l(?) extended the concept of quasiconformal mapping to the Banach space,defined the free quasiconformal mapping(FQC mapping),and studied the geometric characteristics,and obtained a series of results.Thus the basic theory of free quasiconformal mapping in Banach space is established.As we all know,quasisymmetric mapping(QS mapping)is quasiconformal mapping(QC mapping),but QC mapping does not necessarily have quasisymmetry.Therefore,the research on the relationship between QC mapping and QS mapping has been concerned by many mathematicians.In this thesis,we mainly study the relationship between quasiconformal mapping of Banach space and quasisymmetric mapping under the quasihyperbolic metric(QHQS mapping),and thus give an answer to an open problem raised by V(?)is(?)l(?) in 1999 under some specific condition.This thesis consists of four parts,and the specific arrangements are as follows:In Chapter one,we mainly introduce the research background,research status and main results in this thesis.In Chapter two,we introduce the concepts of QC mapping,QS mapping and some basic inequalities.In Chapter three,we explore the open problem proposed by V(?)is(?)l(?) in 1999:Suppose that G and G' are subdomains of E and E' in Banach space,respectively.If the homeomorphic mapping f:G?G' is ?-QHQS mapping,then is f a ?-FQC mapping,where ? depends on the function ??We get that the answer to this question is yes under the condition of Q-regular(Q?1).At the same time,the necessary auxiliary lemma for the main results of this thesis is also proved.In Chapter four,we prove that QHQS mapping is equivalent to FQC mapping under the condition of Q-regular(Q?1).The positive answer of the above open problem under the condition of Q-regular is given.