强John域中的Lipschitz扩张和拟双曲度量

利用拟双曲度量刻划了强John域,并且获得了强John域中拟共形映射的Hardy-Littlewood性质.     

一致域和最大-最小不等式性质

引入区域的最大最小不等式性质,研究最大最小不等式性质和一致域的关系,得到了下述结果: (1)区域的最大最小不等式性质具有拟共形不变性; (2)如果区域D是一致域,则D具有最大最小不等式性质; (3)若D和它的外部D=R2\D具有最大-最小不等式性质,则D是R2中的一致域.     

一致区域与拟双曲度量

设 D 是 R~n(n≥2)的真子区域.F.W.Gehving 与 B.G.Osgood 证明,D 是一致区域的充分必要条件是:存在常数 c 和 d,使得 k_D(x_1,x_2)≤cj_D(x_1,x_2)+d,(?)x_1,x_2∈D.本文证明,这个条件可减弱为:存在一常数 A,使得K_D(x_1,x_2)≤A·j_D(x_1,x_2),(?)x_1,x_2∈D.这里 K_D(x_1,x_2)为 D 中任意两点 x_1,x_2的拟双曲度量,j_D(x_1,x_2)=1/2log([|x_1-x_2|]/[d(x_1,(?)D)]+1)([|x_1-x_2|]/[d(x_2,(?)D)]+1),d(x,(?)D)为 x 到(?)D 的欧氏距离.     

凸域上拟双曲测地线直径的Gehring-Hayman 恒等式

利用型函数和最大项m（σ）的几何意义研究全平面上Dirichlet级数的增长性,得到了Dirichlet级数增长性与系数、指数之间关系的四个结论,推广了以往研究增长性的相关结果.     

距离边界条件域和H(o)lder域

主要利用了距离边界条件域和Holder域的定义,建立了当kD(x1,x2)和jD(x1,x2)满足一定的条件后,两类距离边界条件域和Holder域之间的四个充要条件和一个充分条件.     

拟双曲度量与John圆

设D是R~2中的Jordan域,本文证明了D是b-John圆当且仅当存在常数c≥1,对任意的x_1,x_2∈D,有k_D(x_1,x_2)≤cH_D(x_1,x_2),这里kD(x_1,x_2)表示D中x_1与x_2二点的拟双曲距离,H_D(x_1,x_2)=1/2log(1+(l(γ))/(d(x_1,■D)))(1+(l(γ))/(d(x_2,■D))),其中l(γ)为D中连结x_1与x_2二点的拟双曲测地线的欧几里德长度.     

L(φ,μ)-平均域及其几何特征

本文首先引入L(,μ)-平均域,它是已知的John-域,Ls-平均域和Ls(μ)-平均域等的推广.然后利用拟双曲度量来刻画L(,μ)-平均域的几何特征.最后证明,对于任何满足L(,μ)-平均域定义中条件的函数,John-域是L(,μ)-平均域.     

L(φ,μ)-平均域及其几何特征

本文首先引入 Ｌ（φ,μ）－平均域,它是已知的 Ｊｏｈｎ－域, Ｌｓ－平均域和 Ｌｓ（μ）－平均域等的推广．然后利用拟双曲度量来刻画Ｌ（φ,μ）－平均域的几何特征．最后证明,对于任何满足Ｌ（φ,μ）－平均域定义中条件的函数φ, Ｊｏｈｎ－ 域是 Ｌ（φ,μ）－平均域．     

拟极值距离域和拟共形映照的扩张

本文研究了空间R_n中的拟极值距离城,找到了一个可以拟共等价于球而不是拟极值距离域的例子,给出了拟极值距离域的一个等价定义,证明了一个Jordan的拟极值距离域的外部是拟极值距离城的充分条件为它是拟共形反射域。最后我们还建立了一个平面拟共形映照的扩张性质。     

两类距离边界条件域的拟共不变性和Mobius变换的两个拟不变量

研究了两类距离边界条件域K-Bc和j-Bc,得到了它们的若干性质及它们的关系,证明了在一定条件下K-Bc和j-Bc是拟共不变的以及距离K(x1,x2)和j(x1,x2)在Mobius变换下具有拟不变性.     

度量空间中拟对称映射与拟双曲一致域的研究

本文主要考虑度量空间中拟双曲一致域与拟对称映射之间的关系,并证明了度量空间中拟双曲一致域在拟对称映射下仍然是保持不变的.     

The Apollonian inner metric and uniform domains

In this paper we characterize uniform domains in terms of the Apollonian inner metric and the j‐metric, thus providing solutions to two open problems given in [16]. We also discuss the relationships among uniform, A‐uniform, Apollonian quasiconvex and quasi‐isotropic domains (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrability of superharmonic functions, uniform domains, and Hölder domains

Let denote the space of all positive superharmonic functions on a domain . Lindqvist showed that is a bounded subset of . Using this, we give a characterization of finitely connected -dimensional uniform domains and remarks on Hölder domains.

     

John域

本文研究了Ｊｏｈｎ域，证明了有界一致域必定是Ｊｏｈｎ域和Ｊｏｈｎ域是拟共形不变量．     

The Arc Distortion in QH Inner ψ-uniform （or Convex） Domains in Real Banach Spaces

Let D and D′ be domains in real Banach spaces of dimension at least 2. The main aim of this paper is to study certain arc distortion properties in the quasihyperbolic metric defined in real Banach spaces. In particular, when D′ is a QH inner ψ-uniform domain with ψ being a slow (or a convex domain), we investigate the following: For positive constants c,h,C,M, suppose a homeomorphism f: D → D′ takes each of the 10-neargeodesics in D to (c, h)-solid in D′. Then f is C-coarsely M-Lipschitz in the quasihyperbolic metric. These are generalizations of the corresponding result obtained recently by Väisälä.     

Quantitative Continuous Domains

We relate two approaches to Quantitative Domain Theory, the partial metrics of Steve Matthews and the measurements of Keye Martin, by showing that stable partial metrics are in one-to-one correspondence to weakly modular measurements. It is shown that every -algebraic domain admits a partial metric whose associated measurement assigns zero precisely to the set of maximal elements, a condition which features prominently in Martin's work. A partial metric gives rise to a metric in a standard way; we study the conditions under which the resulting space is complete and show that every -algebraic domain admits a partial metric of this kind. We discuss a number of examples and counterexamples to locate the strength of our results more exactly.     

The Gauge of a Uniform Frame

This note characterizes gauge structures for frames, i.e., families of metric diameters which completely describe frame uniformities. Some applications are presented.     

拟圆的一个充分必要条件

利用拟双曲度量研究了平面拟共形映照中的拟圆，得到了拟圆的一个充分必要条件。