Function theory for Laplace and Dirac-Hodge Operators in hyperbolic space

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC ¹ functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian. Research supported by the National Science Foundation of China (Mathematics Tianyuan Foundation, No A324610) and Hebei Province (105129) Research supported by Academy of Finland     

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy’s type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.

     

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy's type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.     

复Clifford分析中的超单演函数

该文研究复Clifford分析中的超单演函数,即方程z_n Df(z)+(n-1)Qf′=0的解. 记f(z)=Pf(z)+Qf(z)e_n,f(z)∈C^2(Ω),f(z): Ω → C^{n+1},Ω C^{n+1}，得出超单演函数的几个性质.     

Cauchy-type integral formula and Plemelj formula of hypermonogenic functions on unbounded domains

In this paper, we discuss the Cauchy-type integral formula of hypermonogenic functions on unbounded domains in real Clifford analysis, then we extend the Plemelj formula and Cauchy–Pompeiu formula of hypermonogenic functions on bounded domains to unbounded domains. We also deal with the Green-type formula on unbounded domains and get several important corollaries.     

Clifford分析中广义超正则函数的一个非线性边值问题

讨论了Cliffrd分析中广义超正则函数的一个非线性边值问题.首先将广义超正则函数分解为两个奇异积分算子,然后给出了广义超正则函数的Plemelj公式及相关奇异积分算子的性质,最后利用Schauder不动点原理证明了广义超正则函数的一个非线性边值问题的解的存在性及积分表达式.     

广义超正则函数的一个带位移的非线性边值问题

讨论了一个广义超正则函数的带位移的非线性边值问题.首先将这个广义超正则函数分解为两个积分算子的和并讨论了相关奇异积分算子的性质,然后利用超正则函数的Plemelj公式和Schauder不动点定理证明了这个广义超正则函数的带位移的非线性边值问题的解的存在性和唯一性.     

复Clifford分析中几类函数的等价描述

第一部分借助黄沙老师拟置换的思想,得到了复正则函数,复超正则函数的等价条件,定义了复超调和函数并讨论了它的等价条件,第二部分讨论了复超正则函数的若干性质.     

实Clifford分析中超正则函数列和函数空间的性质

定义了实Clifford分析中超正则函数列的一致有界、内闭一致有界及内闭一致收敛等概念,并讨论了超正则函数列及超正则函数空间的几条性质.     

双曲空间中Laplace-Beltrami方程的一个带位移的边值问题

讨论了双曲空间中Laplace-Beltrami方程的一个带位移的边值问题.首先将双曲空间中的Laplace-Beltrami方程的解转化为Clifford分析中的超正则函数.然后给出了超正则函数的Plemelj公式并讨论了相关奇异积分算子的性质,最后利用积分方程的方法和压缩不动点原理证明了Laplace-Beltrami方程的一个带位移的边值问题的解的存在性和唯一性.     

Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C¹.     

k-超正则函数及其相关函数的性质

给出了k -超正则函数的开拓定理和唯一性定理,由唯一性定理证明了超正则函数列的内闭一致收敛性; 由k -超正则函数的P 部和Q 部满足的两个微分方程,讨论了此方程与k -超正则函数及其相关函数的关系.     

Jump problem and removable singularities for monogenic functions

In this article the jump problem for monogenic functions (Clifford holomorphicity) on the boundary of a Jordan domain in Euclidean spaces is investigated. We shall establish some criteria that imply the uniqueness of the solution in terms of a natural analogue of removable singularities in the plane to ℝⁿ⁺¹ (n ≥ 2). Sufficient conditions to extend monogenically continuous Clifford algebra valued functions across a hypersurface are proved. Communicated by Jenny Harrison     

Meromorphic functions sharing four small functions on 2CM + 2IM

In this paper, we prove a result that if two meromorphic functions share two small functions CM^* and two other small functions IM^*, then the two meromorphic functions share those four distinct small functions CM^*, and give more detailed results. Which generalizes G. G. Gundersens 2CM+2IM Theorem from sharing values to sharing small functions in detailed form.Received: 5 November 2002     

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

The implicit function theorem in a non-Archimedean setting

In this paper, the inverse function theorem and the implicit function theorem in a non-Archimedean setting will be discussed. We denote by N any non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from Nⁿ to N^m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from Nⁿ to Nⁿ and the implicit function theorem for functions from Nⁿ to N^m with m<n.     

A characterization of Fourier series of Stepanov almost-periodic functions

We show by example that the classical characterization of the Fourier series of periodic functions in L^p, 1<p≤+∞, as those trigonometric series whose Abel or Fejér means are uniformly bounded in L^p does not hold for general (non-periodic) trigonometric series in relation to Stepanov-almost-periodic functions, but that it does hold under the additional hypothesis that the means are translation equicontinuous. We exhibit a bounded, infinitely differentiable function that belongs to every class of Besicovitch-almost-periodic functions but is not equivalent in the metric of Besicovitch-almost-periodic functions to any Stepanov-almost-periodic function.     

On functions convex in the direction of the real axis with a fixed second coefficient

In the paper we consider the class Γ of analytic and univalent functions f in the unit disk Δ, normalized by f(0) = f′(0) − 1 = 0, having real coefficients and such that f(Δ) is convex in the direction of the real axis. We are especially interested in some subclasses of Γ. The most important of them is Γ(c) consisting of those functions which have the second coefficients of the Taylor expansion fixed and equal to c. We obtain the Koebe set for this class as well as for the classes Γ⁺(c) and Γ⁻(c) of functions which are in some sense convex in the direction of positive and negative axes respectively.