Some Properties Of Quasisymmetric Mappings And John Domains

As a significant research object in complex analysis,analytic functions have a lot of useful properties,such as uniqueness,integrals being independent of related curves,extension and so on.As a generalization of complex analytic functions,quasiconformal maps were proposed by Grotsch in 1928 while he studied Riemann mapping theo-rem.After that Quasiconformal mappings attract much attention.Beurling and Ahlfors' research shows that the restriction of quasiconformal mappings on the upper plane to the real axis posses many significant properties.This class of mappings is called quasisymmetric mappings.The general concept for quasisymmetric mappings was posed by Tukia and Vais(?)l(?) in 1980.After its appearance,quasisymmetric mappings become an important research object.Obviously,quaisysmmetric mappings are quasiconformal,and so they are also generalizations of analytic functions.A natural problem is whether quasisym-metric mappings have extension property similar to that for analytic functions.This is our first study problem in this thesis.In order to investigate the elasticity and injectivity theory,John introduced John domains in 1961.Since then John domains have become a hot research object.Vais(?)l(?) discussed the extension property of John domains in Rn,which has been generalized to the case of Banach spaces later.?-John domains are a generalization of John domains.Then naturally one asks whether the extension property of John domains is still valid for ?-John domains.This is the second study problem in this thesis.To study the approximation and injectivity problems,in 1978,Martio and Sarvas introduced a class of domains which are named as uniform domains.According to the hyperbolic characterization of uniform domains,Vuorinen in-troduced a new class of domains called ?-uniforms domain in 1985.In 2005,Hasto,Klen,Sahoo and Vuorinen raised an open problem when they investi-gated the geometric properties of ?-uniform domains.This open problem is the third study problem in this thesis.This thesis is composed of the discussions of these three problems as men-tioned above.It consists of four chapters,and the concrete arrangement is as follows.In Chapter one,we provide the background on the studied problems and the statement of our main results.In Chapter two,we investigate the first question,that is,the extension of quasisymmetric mappings in Banach spaces.We prove that the mapping which is quasisymmetric on two subsets,respectively,is still quasisymmetric under certain conditions.This conclusion generalizes the corresponding result of Vais(?)l(?) in Ann.Acad.Sci.Fenn.Math.in 1991.In Chapter three,we discuss the second question,that is,the the exten-sion property of ?-John domains.We demonstrate that the union of two ?-John domains is still ?'-John domain under certain conditions.This assertion gen-eralizes the corresponding result of Li and Wang published in Ann.Acad.Sci.Fenn.Math.in 2010.In Chapter four,we study the open problem raised by Hasto,Klen,Sahoo and Vuorinen in 2005,i.e.,the third study problem in this thesis.By construct-ing a counterexample,we give a negative answer to this open problem.