This paper studies the quasisymmetric mappings on Moran sets. We introduce a gener- alized form of weak quasisymmetry and prove that, on Moran set satisfying the small gap condition, a generalized weakly quasisymmetri...This paper studies the quasisymmetric mappings on Moran sets. We introduce a gener- alized form of weak quasisymmetry and prove that, on Moran set satisfying the small gap condition, a generalized weakly quasisymmetric mapping is quasisymmetric. We further give a criterion for the quasisymmetry of mappings between Moran sets with some regular structure.展开更多
It is proved that Kq(h)=K0(h) for every h in some class of quasisymmetric mappings of the unit circle with substantial points, where Kq(h):=sup{M(h(Q))/M(Q); Qis a quadrilateral with the domain unit disk} and K0(h) is...It is proved that Kq(h)=K0(h) for every h in some class of quasisymmetric mappings of the unit circle with substantial points, where Kq(h):=sup{M(h(Q))/M(Q); Qis a quadrilateral with the domain unit disk} and K0(h) is the extremal maximum dilatation of h.展开更多
The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrig...The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.展开更多
The aim of this paper is to investigate the relationship between relative quasisymmetry and quasimöbius in quasi-metric spaces,and show that a homeomorphism f isη-quasisymmetric relative to A if and only if it i...The aim of this paper is to investigate the relationship between relative quasisymmetry and quasimöbius in quasi-metric spaces,and show that a homeomorphism f isη-quasisymmetric relative to A if and only if it isθ-quasimöbius relative to A between two both bounded quasi-metric spaces,where A⊆X and X is a quasi-metric space.展开更多
Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal con...Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal conformal modulus dilatation of the boundary homeomorphism h, K 1(h) is the maximal dilatation of extremal quasiconformal mappings that agree with h on the boundary. In addition, when the complex dilatation of an extremal quasiconformal mapping is known, the proof of the result simplifies Reich and Chen Jixiu-Chen Zhiguo’s result.展开更多
The Beurling-Ahlfors’ extension is studied under relatively general conditions and its dilatation fonction is estimated. Particularly, the classic Deurling -Ahlfors’ theorem can be obtained under the M-condition.
基金Supported by NSFC(Grant Nos.11071224,11201155)Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications
文摘This paper studies the quasisymmetric mappings on Moran sets. We introduce a gener- alized form of weak quasisymmetry and prove that, on Moran set satisfying the small gap condition, a generalized weakly quasisymmetric mapping is quasisymmetric. We further give a criterion for the quasisymmetry of mappings between Moran sets with some regular structure.
文摘It is proved that Kq(h)=K0(h) for every h in some class of quasisymmetric mappings of the unit circle with substantial points, where Kq(h):=sup{M(h(Q))/M(Q); Qis a quadrilateral with the domain unit disk} and K0(h) is the extremal maximum dilatation of h.
基金Supported by the National Natural Science Foundation of China(10671174, 10401036)a Foundation for the Author of National Excellent Doctoral Dissertation of China(200518)
文摘The relationship between Strebel boundary dilatation of a quasisymmetric function h of the unit circle and the dilatation indicated by the change in the modules of the quadrilaterals with vertices on the circle intrigues many mathematicians. It had been a conjecture for some time that the dilatations Ko（h） and K1（h） of h are equal before Anderson and Hinkkanen disproved this by constructing concrete counterexamples. The independent work of Wu and of Yang completely characterizes the condition for Ko（h） = K1 （h） when h has no substantial boundary point. In this paper, we give a necessary and sufficient condition to determine the equality for h admitting a substantial boundary point.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671057,11901136)the Guizhou Provincial Science and Technology Foundation(Grant No.[2020]1Y003)the Ph D research startup foundation of Guizhou Normal University(Grant No.11904/0517078)。
文摘The aim of this paper is to investigate the relationship between relative quasisymmetry and quasimöbius in quasi-metric spaces,and show that a homeomorphism f isη-quasisymmetric relative to A if and only if it isθ-quasimöbius relative to A between two both bounded quasi-metric spaces,where A⊆X and X is a quasi-metric space.
文摘Two necessary and sufficient conditions for the validity of the conjecture K 0(h)=K 1(h) are given, which are independent of the complex dilatations of extremal quasiconformal mappings, where K 0(h) is the maximal conformal modulus dilatation of the boundary homeomorphism h, K 1(h) is the maximal dilatation of extremal quasiconformal mappings that agree with h on the boundary. In addition, when the complex dilatation of an extremal quasiconformal mapping is known, the proof of the result simplifies Reich and Chen Jixiu-Chen Zhiguo’s result.
文摘The Beurling-Ahlfors’ extension is studied under relatively general conditions and its dilatation fonction is estimated. Particularly, the classic Deurling -Ahlfors’ theorem can be obtained under the M-condition.
