CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

Plemelj Formula for Cauchy Type Integral on Certain Distinguished Boundary in Universal Clifford Analysis

Under the foundation of Cauchy integral formula on certain distinguished boundary for functions with values in universal Clifford algebra, we define the Cauchy type integral with values in a universal Clifford algebra, obtain its Cauchy principal value and Plemelj formula on certain distinguished boundary.

THE GAUSS–GREEN THEOREM IN CLIFFORD ANALYSIS AND ITS APPLICATIONS

In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.

SPHERICAL MEANS, DISTRIBUTIONS AND CONVOLUTION OPERATORS IN CLIFFORD ANALYSIS

New higher dimensional distributions are introduced in the framework of Clifford analysis.They complete the picture already established in previous work, offering unity and structuralclarity. Amongst them are the building blocks of the principal value distribution, involvingspherical harmonics, considered by Horvath and Stein.

The Teodorescu Operator in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators and which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator __0. In this paper, Teodorescu operators for the Hermitian Dirac operators c9~_ and 0_~, are constructed. Moreover, the structure of the Euclidean and Hermitian Teodor- escu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators is- suing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed.

Multi-vector Spherical Monogenics,Spherical Means and Distributions in Clifford Analysis

New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂finite part＂ distribution Fp x＋^μ on the real line and the generalized spherical means involving vector-valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector-valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self-contained, it offers an overview of the results found so far.

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^∞（R^4n W4n） →4 C^∞（R^4n, W4n）, where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B, the bicomplex number B, and the Clifford algebra R0,4n. The operator D is a square root of the Laplacian in R^4n, introduced by the formula D = ∑j=0^3=0 Kjδzj with Kj being the basis of B, and δzj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B×R0,4n whose definition involves a delicate construction of the bicomplex Witt 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.

HILBERT-DIRAC OPERATORS IN CLIFFORD ANALYSIS

Around the central theme of 'square root' of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.

On the Fourier Spectra of Distributions in Clifford Analysis

In recent papers by Brackx, Delanghe and Sommen, some fundamental higher dimensional distributions have been reconsidered in the framework of Clifford analysis, eventually leading to the introduction of four broad classes of new distributions in Euclidean space. In the current paper we continue the in-depth study of these distributions, more specifically the study of their behaviour in frequency space, thus extending classical results of harmonic analysis.

Hlder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for H61der continuous circulant （2 × 2） matrix functions, on the d- summable （or fractal） boundary F of a Jordan domain Ω in R2n, has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the HSlder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the HSlder exponents, the diameter of F and a specific d-sum （d 〉 d） of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.

Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis

We present recent work of harmonic and signal analysis based on the complex Hardy space approach.

Boundary value problems for two types of degenerate elliptic systems in R^4

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system＇s solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.

SZEG KERNEL FOR HARDY SPACE OF MATRIX FUNCTIONS

By the characterization of the matrix Hilbert transform in the Hermitian Clifford analysis, we introduce the matrix Szeg5 projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded sub-domain of even dimensional Euclidean space, establish the Kerzman-Stein formula which closely connects the matrix Szego projection operator with the Hardy projection operator onto the Hardy space, and get the matrix Szego projection operator in terms of the Hardy projection operator and its adjoint. Furthermore, we construct the explicit matrix Szego kernel function for the Hardy space on the sphere as an example, and get the solution to a boundary value problem for matrix functions.

The Mehler Formula for the Generalized Clifford-Hermite Polynomials

The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler＇s formula for the generalized Clifford-Hermite polynomials of Clifford analysis.

Real Clifford Windowed Fourier Transform

We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier transform （CWFT）. Based on the spectral representation of the Clifford Fourier transform （CFT）, we derive several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an example to show the differences between the classical windowed Fourier transform （WFT） and the CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWFT.

Almansi-Type Decomposition Theorem for Bi-k-regular Functions in the Clifford Algebra Cl_(2n+2,0)(R)

Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domainΩ⊆R^(n+1)×R^(n+1)centered at the origin with values in the Clifford algebra Cl_(2n+2,0)(R)is proved.As a corollary,Almansi-type decomposition theorem for biharmonic functions of degree k is given.

Some Generalized Clifford-Jacobi Polynomials and Associated Spheroidal Wavelets

In the present paper,by extending some fractional calculus to the framework of Clifford analysis,new classes of wavelet functions are presented.Firstly,some classes of monogenic polynomials are provided based on 2-parameters weight functions which extend the classical Jacobi ones in the context of Clifford analysis.The discovered polynomial sets are next applied to introduce new wavelet functions.Reconstruction formula as well as Fourier-Plancherel rules have been proved.The main tool reposes on the extension of fractional derivatives,fractional integrals and fractional Fourier transforms to Clifford analysis.

A Kind of Boundary Value Problem for Hypermonogenic Function Vectors

By the Plemelj formula and the compressed fixed point theorem,this paper discusses a kind of boundary value problem for hypermonogenic function vectors in Clifford analysis.And the paper proves the existence and uniqueness of the solution to the boundary value problem for hypermonogenic function vectors in Clifford analysis.

The Fixed Point Theorem and the Iterative Approximation of Modified Cauchy Integral Operator of Regular Functions

In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T{f}‖α and ‖f‖α. In the second part of this paper, we introduce the modified cauchy integral operator T^- for regular functions. Firstly, we prove that the operator T^- has a unique fixed point by the Banach＇s Contraction Mapping Principle. Secondly, we give the Mann iterative sequence, and then we show the iterative sequence strongly converges to the fixed point of the operator T^-.

Hyperbolic L_2-modules with Reproducing Kernels

In this paper, the Dirac operator on the Klein model for the hyperbolic space is considered. A function space containing L2-functions on the sphere S^m-1 in R^m, which are boundary values of solutions for this operator, is defined, and it is proved that this gives rise to a Hilbert module with a reproducing kernel.