泛Clifford分析中无界域上的Cauchy积分公式和Cauchy-Pompeiu公式

本文研究了泛Clifford分析中的Cauchy积分公式和Cauchy-Pompeiu公式.通过引入修正的Cauchy核,得出了取值在泛Clifford代数上的两公式在无界域上的表达式.此两公式是有界域上的相应结果的推广,并为研究无界域上的边值问题打下了基础.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in Cn given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in Cn, which is different from the well-known Leray formula, This new formula eliminates the term that contains the parameter λ from the classical Leray formula, and especially on some domains the uniform estimates for the -equation are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in Cn, which are different from the classical ones, when we properly select the vector function W.     

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.

     

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.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

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.     

Integral Formula of Isotonic Functions over Unbounded Domain in Clifford Analysis

In this paper we mainly study the so-called isotonic Dirac system over more general types of unbounded domains in Euclidean space of even dimension. In such systems different Dirac operators in the half dimension act from the left and from the right on the functions considered. We obtain the integral representation of isotonic functions satisfying the decay condition over the unbounded domains, and show that the integral representation formula over the unbounded domains for holomorphic functions of several complex variables and for Hermitean monogenic functions may be derived from it.     

A Martinelli-Bochner formula on fractal domains

A new perspective on a Cauchy integral formula for Clifford algebras valued functions on domains with quite smooth boundaries was discussed in [5]. On the other hand, the Cauchy transform associated to Clifford analysis has been involved recently with fractional metric dimensions and fractals, see [1, 2, 3]. In this paper we consider the question of possible generalizations of the Cauchy integral formula to domains with fractal boundary. As an application, we prove a Martinelli-Bochner type formula for several complex variables on such pathological domains. The proof makes heavy use of the isotonic approach of the monogenic functions theory. Received: 8 October 2008     

On the Hadamard formula for nonsmooth domains

We consider the first eigenvalue of the Dirichlet-Laplacian in three cases: C1,1-domains, Lipschitz domains, and bounded domains without any smoothness assumptions. Asymptotic formula for this eigenvalue is derived when domain subject arbitrary perturbations. For Lipschitz and arbitrary nonsmooth domains, the leading term in the asymptotic representation distinguishes from that in the Hardamard formula valid for smooth perturbations of smooth domains. For asymptotic analysis we propose and prove an abstract theorem demonstrating how eigenvalues vary under perturbations of both operator in Hilbert space and Hilbert space itself. This abstract theorem is of independent interest and has substantially broader field of applications.     

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     

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.     

Hobson's formula for Dunkl operators and its applications

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials.     

An Improved Cauchy Formula for Hypermonogenic Functions

A new Cauchy-type formula for hypermonogenic functions is derived. Hypermonogenic functions, introduced in [6], are a generalization of holomorphic functions to several dimensions. The power function x^m is hypermonogenic. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš     

Geometric estimates for the trace formula

In order to study the asymptotic distribution of geometric or spectral data associated with quotients of a reductive group by a lattice, one needs a trace formula for test functions on that group with noncompact support. Arthur has proved a trace formula for compactly supported test functions on reductive groups of arbitrary rank. We show that the coarse geometric expansion in his formula converges for rapidly decreasing functions.      

Generalization of the Martinelli-Bochner integral representation

The well known Martinelli-Bochner integral formula from the theory of holomorphic functions of many variables is extended to arbitrary unbounded domains.     

Stable local trace formula

Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.     

Applications of a formula on Beltrami flow

In this note, we obtain uniqueness results for Beltrami flow in both bounded and unbounded domain with nonempty boundary by establishing an elementary but useful formula involving operators div and curl. We also use this formula to deal with Maxwell and Stokes eigenvalue problems.     

From Tanaka's formula to Ito's formula: The fundamental theorem of stochastic calculus

In this article we give a new proof of Ito's formula inR ⁿ starting from the one-dimensional Tanaka formula. The proof is algebraic and does not use any limiting procedure. It uses the integration by parts formula, Fubini's theorem for stochastic integrals and essential properties of local times.     

A positive radial product formula for the Dunkl kernel

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.