Studies On The Quasisymmetry Of Quasiconformal Mappings In Loewner Spaces

In the 80s of last century,Tukia and Vaisala gave the general definition of quasisymmetric mapping,which has been an important research object since its inception.It is well known,quasisymmetric mapping must be weakly quasi-symmetric mapping,but the converse is not true.Many scholars have studied the equivalence between weakly quasisymmetric mapping and quasisymmetric mapping.The main content of this thesis is to prove that weak quasisymmetric map-ping in Loewner space is actually quasisymmetric under certain conditions.The paper consists of four Chapters,the specific contents are as follows:In Chapter 1,we introduce the background of the research problem and the main results obtained,and emphasize on the concept and properties of Q-regular space and quasisymmetric mapping.In Chapter 2,we introduce the related preparatory knowledge,including the concepts of module of curve family,linear locally connected set and some properties of them.In Chapter 3,we discuss some properties in the Q-regular space,which play a key role in the proof of this thesis.In Chapter 4,we show that weakly quasisymmetric mappings in Loewner Spaces are actually quasisymmetric under certain conditions.