半空间中一类推广的Poisson积分的增长性质

本文研究了推广的Poisson积分的增长性问题.利用复平面中经典的Hayman定理及其证明方法,通过修改上半空间中的Poisson积分,获得了上半空间中一类位势在无穷远点的增长性质,推广了Hayman定理在高维空间的结果.     

锥中修改Green函数的构造及其应用

首先构造了一个锥中修改的Green函数,给出了其增长性质.作为应用,接着得到了锥中Dirichlet问题的通解.最后证明了这种解表示形式的唯一性.     

半空间中一类调和函数的例外集

利用Whitney方体的相关性质, 给出了一类调和函数在半空间中无穷远点处的增长估计, 且刻画了其例外集的几何性质. 本文推广了张艳慧和邓冠铁在半空间中的相关结果.     

微分多项式的例外集

设ｆ为任一超越整函数,为ｆ的任一微分多项式;本文证明了在满足｜ａ＿ｎ－ａ＿ｍ｜＞ε｜ａ＿ｎ｜（ｎ≠ｍ）和的无穷个圆盘的并集之外取任意有限非零复数无穷次。     

亚纯函数导数的例外集

1 引言及结果 设f是复平面C中超越亚纯函数.亚纯函数a_i(z)称为小函数,若a_i(z)满足T(r,a_i)=o(T(r,f))(i=1,2,…)。我们采用Nevanlinna理论中常用记号,用S(r,f)表示量:当f为有穷级时S(r,f)=O(log r);当f为无穷级时S(r,f)=O(log r T(r,f)),至多除去r的一个有限测度集。     

亚纯函数的例外集

自从Ｌｅｈｔｏ Ｏ．［１］提出例外集的概念后,这方面已有不少工作,但这些工作基本上是局限于整函数来做的,主要原因是极点给证明带来很大困难．本文证明了任一个满足δ（∞,ｆ）＞３／４的超越亚纯函数ｆ（ｚ）,ｆｋＱ［ｆ］在任一不含ｆ和Ｑ［ｆ］极点的可数个圆盘并集之外取任何非零复数无穷次,其中ｋ≥１５／８＋１／２ГＱ,ГＱ是ｆ的微分多项式 Ｑ［ｆ］的权．本文的结果推广了 Ｈａｘｍａｎ等人的结论．     

平面中Poisson方程的Dirichlet问题

基于向量旋转内积不变的特点,通过对Green积分公式的推广,得到与空间Green三个公式相似的平面Green公式.从而,得到平面中Poisson方程Robin问题的解和平面中Poisson方程Dirichlet问题的解.     

上半空间中一类次调和函数的增长估计

本文证明了n-维(n≥2)Euclidean空间的上半空间中Poisson积分在无穷远点处的增长性质.同时将这个性质推广到次调和函数中去,其概括了解析函数和调和函数的增长性质.     

亚纯函数的例外集

亚纯函数的例外集问题的已有结论,还未触及例外集内含有极点的情形.本文证明了对于满足δ(∞,f)>0的超越亚纯函数f(z),设F=fk则F′的可数个圆盘并集之外取任何非零有穷复数无穷次,或者取∞无穷次,本文推广了Hayman,Andersom等人的结论.     

球面上的Poisson积分

证明了球面上的Poisson积分算子从Lp(Sn?1)到Lorentz空间Lq,1(B1)(q 1)有界,且从有界Borel测度集M(Sn?1)到Lq,1(B1)(q < nn?1)有界,推广了部分已知的结果.进一步构造了一个反例说明了球面上的Poisson积分算子不一定从M(Sn?1)到L n n?1(B1)有界.     

Comparison of Green Functions and Harmonic Measures for Parabolic Operators

In this paper, we prove two-sided pointwise estimates for the Green function of a parabolic operator with singular first order term on a C^1,1-cylindrical domain . Basing on these estimates, we establish the equivalence of the parabolic measure, the adjoint parabolic measure and the surface measure on the lateral boundary of . These results are first studied by some authors for certain elliptic and less general parabolic operators. Mathematics Subject Classifications (2000) 31B25, 35B05, 35K10, 58J35.     

Poisson Kernel and Green Function of the Ball in Real Hyperbolic Spaces

Let (X _t ) _t⩾0 be the n-dimensional hyperbolic Brownian motion, that is the diffusion on the real hyperbolic space having the Laplace–Beltrami operator as its generator. The aim of the paper is to derive the formulas for the Gegenbauer transform of the Poisson kernel and the Green function of the ball for the process (X _t ) _t⩾0. Under additional hypotheses we prove integral representations for the Poisson kernel. This yields explicit formulas in and spaces for the Poisson kernel and the Green function as well.      

Estimates of Green Function for Relativistic α-Stable Process

We call an R ^d -valued stochastic process X _t with characteristic function exp{–t{(m ^2/+²)^/2–m}},R ^d ,m>0, the relativistic -stable process. In the paper we derive sharp estimates for the Green function of the relativistic -stable process on C ^1,1 domains. Using these estimates we provide lower and upper bounds for the Poisson kernel. As another application we derive 3G Theorem and Boundary Harnack Principle for C ^1,1 domains.     

亚纯函数微分多项式f~kQ[f]+P[f]的零点分布

从例外集的角度研究了亚纯函数微分多项式fkQ[f]+P[f]的零点分布,证明了:对于满足δ(∞,f)1-α>0的超越亚纯函数f(z),微分多项式fkQ[f]+P[f]在不含极点的可数个圆盘并集之外有无穷多个零点,其中k>1+ΓP+γP+α1(-1+αΓQ+ΓP-γP),ΓQ是Q[f]的权,ΓP,γP是P[f]的权和次数.本文推广了Hayman,Anderson,Langley等人的结论.     

Bergman kernel on generalized exceptional hua domain

We have computed the Bergman kernel functions explicitly for two types of generalized exceptional Hua domains, and also studied the asymptotic behavior of the Bergman kernel function of exceptional Hua domain near boundary points, based on Appell's multivariable hypergeometric function.     

On the Size of the Exceptional Set in Nevanlinna's Second Fundamental Theorem for Certain Classes of Meromorphic Functions

Some conditions on the size of the exceptional set that arise in Nevanlinna's Second Fundamental Theorem are established, showing that previous sharp results can be improved by restricting the class of functions considered and suggesting a close relationship between the size of the exceptional set and the lower growth of the characteristic function. Examples of functions of rapid growth possessing exceptional sets are built, showing that these conditions are sharp for the class of functions considered.     

Green functions and related boundary value problems on the Heisenberg group

In this article, a modified Kelvin transform on ? _n using inversion with respect to a ball of arbitrary radius is defined, which gives explicit expressions for Green's function and Poisson's kernel for the Korányi ball of arbitrary radius and annular domain. The solution of the Dirichlet problem for the union of two balls is discussed using the Schwarz's alternating method.     

Complexity in Complex Analysis

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable. Such domains turn out to have the property that their classical domain functions all extend to be meromorphic functions on a compact Riemann surface, and this condition will be shown to be equivalent to the condition that an Ahlfors map and its derivative are algebraically dependent. We also show how many of these results can be generalized to finite Riemann surfaces.     

Approximation of Subharmonic Functions in the Half-Plane by the Logarithm of the Modulus of an Analytic Function

We approximate subharmonic functions defined in the open half-plane in the uniform metric outside the exceptional set by the logarithm of the modulus of an analytic (in the half-plane) function for the cases of a finite order and of an infinite lower order. We also obtain an estimate for the size of the exceptional set. It is shown that, in the case of a finite order, the obtained accuracy of the approximation cannot be essentially improved.