Clifford分析中Cauchy型奇异积分算子的一些性质

First,we give a module estimation of the singular integral with a differential element.Then by proving the existences of Cauchy principal value we obtain the transformation formula of the Cauchy-type singular integrals with a parameter.     

Clifford分析中奇异积分的Poincaré-Bertrand置换公式

借助于多元复分析的思想，本文证明了Ｃｌｉｆｏｒｄ分析中奇异积分的Ｐｏｉｎｃａｒé－Ｂｅｒｔｒａｎｄ置换公式．     

Clifford分析中对偶的k-Hypergenic函数

研究了取值于实Clifford代数空间Cl_(n+1,0)(R)中对偶的k-hypergenic函数.首先,给出了对偶的k-hypergenic函数的一些等价条件,其中包括广义的Cauchy-Riemann方程.其次,给出了对偶的hypergenic函数的Cauchy积分公式,并且应用其证明了(1-n)-hypergenic函数的Cauchy积分公式.最后,证明了对偶的hypergenic函数的Cauchy积分公式右端的积分是U\Ω_2中对偶的hypergenic函数.     

实Clifford分析中的拟Bochner Martinelli型高阶奇异 积分

该文借助于高阶奇异积分的Hadmard主值思想以及归纳法思想讨论了实Clifford分析中拟Bochner Martinelli型高阶奇异积分Hadmard主值的存在性、递推公式、计算公式,以及在Hadamard主值意义下的微分公式.     

泛Clifford分析中的Laurent展式和留数定理

该文由泛Clifford分析中在特异边界上的Cauchy积分式得出了具有孤立奇点的LR正则函数在其相应的Laurent域上的Laurent展式，并由此给出了留数的定义，得出了类似于经典函数理论的留数定理。     

Clifford分析中两个Cauchy型奇异积分算子的换序公式

先证明了Clifford分析中普通积分算子在Liapunov闭曲面上的换序公式,然后证明了两个Cauchy型奇异积分算子的存在性及性质,最后给出了这两个奇异积分算子的换序公式.     

On the Cauchy Principal Value of the Khenkin-Ramirez Singular Integral in Strictly Pseudoconvex Domains of ℂ <Superscript>n</Superscript>

We show that in the multidimensional case (unlike the complex plane) the Cauchy principal value of the Khenkin-Ramirez singular integral in strictly pseudoconvex domains is equal to the limit value of this integral inside the domain.Original Russian Text Copyright © 2005 Kytmanov A. M. and Myslivets S. G.The first author was supported by a grant of the President of the Russian Federation and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1212.2003.1); the second author was supported by the Krasnoyarsk Region Science Foundation (Grant 12F0063C).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 625–633, May–June, 2005.     

Bicomplex Hermitian Clifford analysis

Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator |D:C∞(R4n,W4n)→C∞(R4n,W4n), where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B^, the bicomplex number B, and the Clifford algebra Rn. The operator D is a square root of the Laplacian in R4n, introduced by the formula D|=∑j=03Kj?Zj with Kjbeing the basis of B^, and ?Zj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B?R0,4n whose definition involves a delicate construction of the bicomplexWitt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.     

Clifford分析中k-正则函数的一些性质和高阶Cauchy型积分的边值特性

本文首先给出了定义于R~n取值于Clifford代数C(V_(n,0))中k-正则函数的若干性质,如唯一性定理,Cauchy-Pompeiu公式,高阶Cauchy积分公式,平均值定理等,然后在k-正则函数的高阶Cauchy积分公式的基础上,相应的定义了r次连续可微函数的高阶Cauchy型积分,并给出了它的Cauchy主值,Plemelj公式,边值的Ho|¨lder连续性及其Privalov定理.     

Clifford分析中双正则函数的Taylor展式及其性质

首先借助实Clifford分析中双正则函数的累次积分的换序公式,给出了双正则函数的Cauchy积分公式,然后由特征边界的Cauchy积分公式,得到了双正则函数的Taylor展式,并由此给出了双正则函数的唯一性定理,柯西不等式和Weierstrass定理.     

实Clifford分析中含两个奇点的拟Bochner-Martinelli型高阶奇异积分

在实C lifford分析中讨论了带有两个奇点的拟Bochner-M artinelli型高阶奇异积分的归纳定义,H adm ard主值的存在性,递推公式,计算公式,以及在H adam ard主值意义下的微分公式.     

Clifford分析中Isotonic柯西型积分的边界性质

本文主要刻画了定义于偶数维欧氏空间中光滑曲面而取值于复Clifford代数的isotonic柯西型积分的边界性质.对具有H(o|¨)lder密度函数的isotonic柯西型积分,得到了Privalov定理和Sokhotskii-Plemelj公式,并证明了多复变函数论中经典Bochner-Martinelli型积分的Privalov定理和Sokhotskii-Plemelj公式为其特殊情形.     

A matrix Hilbert transform in Hermitean Clifford analysis

Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant (2×2) matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis.     

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.     

On the Riemann Hilbert type problems in Clifford analysis

In this paper boundary value problems combining Jump — Riemann and Hilbert problems for monogenic functions in Ahlfors-David regular surfaces and in the upper half space respectively are investigated. The explicit formula of the solution is obtained.     

无界区域上Cauchy型奇异积分算子的性质

On the basis of introducing the modified Cauchy kernel, we discuss the Hoelder continuity of the Cauchy-type singular integral operator on unbounded domains for regular functions by dividing into the following three cases： two points are on the boundary of region; one point is on the boundary and another point is in the interior（or exterior） of the region; two points are in the interior （or exterior） of the region.     

Clifford分析中于特征流形上奇异积分方程的正则化

借助于多元复分析的思想，此文利用Clifford分析中于特征流形上奇异积分的两种Poincare＇-Bretrand置换公式，研究特征流形上奇异积分方程的Fredhlom理论，找到了它的正则化算子．     

Modified Gauss rules for approximate calculation of some strongly singular integrals

The approach we follow consists in transforming the numerical evaluation of hyper-singular integrals into the calculation of a nearly singular integral whose mass is distributed according to a positive parameter ε. To evaluate the latter we apply a Gauss quadrature formula associated with a nearly singular weight function. It is estimated the error in terms of ε. Some numerical results are presented.     

一种Cauchy型主值积分的求积公式的一致收敛性

该文首先给出Cauchy型主值积分φ(wf,x)的一种求积公式φ_m^*(wf,x),然后证明序列$φ_m^*(wf,x)}_m=2^∞在整个闭区间[-1,1]上是一致收敛到Cauchy型主值积分φ(wf,x)的,同时给出它的误差界.