Cauchy Integral Formulae in Quaternionic Hermitean Clifford Analysis

The theory of complex Hermitean Clifford analysis was developed recently as a refinement of Euclidean Clifford analysis; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean Dirac operators constituting a splitting of the traditional Dirac operator. In this function theory, the fundamental integral representation formulae, such as the Borel?CPompeiu and the Clifford?CCauchy formula have been obtained by using a (2 ×?2) circulant matrix formulation. In the meantime, the basic setting has been established for so-called quaternionic Hermitean Clifford analysis, a theory centred around the simultaneous null solutions, called q-Hermitean monogenic functions, of four Hermitean Dirac operators in a quaternionic Clifford algebra setting. In this paper we address the problem of establishing a quaternionic Hermitean Clifford?CCauchy integral formula, by following a (4?× 4) circulant matrix approach.     

Martinelli-Bochner formula using Clifford analysis

In the even dimensional case the Dirac equation may be reduced to the so-called isotonic Dirac system, in which different Dirac operators appear from both sides in half the dimension. This system is then used to derive the classical Martinelli-Bochner formula for several complex variables. Frank Sommen: Supported by FWO-Krediet aan Navorsers 1.5.065.04. Dixan Pe?a Pe?a: Supported by a Doctoral Grant of the Special Research Fund of Ghent University. Received: 8 March 2006     

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.     

Taylor Series in Hermitean Clifford Analysis

In this paper, we consider the Taylor decomposition for h-monogenic functions in Hermitean Clifford analysis. The latter is to be considered as a refinement of the classical orthogonal function theory, in which the structure group underlying the equations is reduced from mathfrakso(2m){mathfrak{so}(2m)}to the unitary Lie algebra u(m).     

Gel'fand–Tsetlin bases of orthogonal polynomials in Hermitean Clifford analysis

An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centered around the simultaneous null solutions of two Hermitean conjugate complex Dirac operators. Copyright © 2011 John Wiley & Sons, Ltd.     

Two integral operators in Clifford analysis

Segal-Bargmann space F₂(Cn) and monogenic Fock space M₂(Rn+1) are introduced first. Then, with the help of exponential functions in Clifford analysis, two integral operators are defined to connect F₂(Cn) and M₂(Rn+1) together. The corresponding integral properties are studied in detail.     

Cauchy transform on nonrectifiable surfaces in Clifford analysis

The problem of reconstructing a monogenic Clifford algebra valued function on the boundary Γ of a general open set Ω in Rn+1 from a prescribed jump data u over the boundary is deeply connected with the study of the Clifford-Cauchy transform

     

Commutators and singular integral operators in Clifford analysis

In this paper we develop a method for setting the compactness of the commutator relative to the singular integral operator acting on Hölder continuous functions over Ahlfors David regular surfaces in R ⁿ⁺¹ . This method is based on the essential use of the monogenic decomposition of Hölder continuous functions. We also set forth explicit representations of the adjoints of the singular Cauchy type integral operators, relative to a total subset of real functionals.     

On the existence of approximate solutions for singular integral equations of Cauchy type discretized by Gauss-Chebyshev quadrature formulae

It is shown that the direct Gauss-Chebyshev method used for the numerical solution of singular integral equations of Cauchy-type possesses a unique solution for sufficiently largen.     

A Paley-Wiener theorem and Wiener-Hopf-type integral equations in Clifford analysis

Using the properties of the monogenic extension of the Fourier transform, we state a Paley-Wiener-type theorem for monogenic functions. Based on an multiplier algebra related to boundary values of monogenic functions we consider integral equations of Wiener-Hopf-typeK±u _±=f on ℝ ⁿ , whereK ∈S′ andu _± are boundary values of monogenic functions in ℝ₊ ⁿ⁺¹ and ℝ_{_} ⁿ⁺¹ respectivly.     

Holder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for H?lder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ?²ⁿ , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the H?lder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the H?lder exponents, the diameter of Γ 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.     

An integral transform generating elementary functions in Clifford analysis

In this paper we introduce a real integral transform which links trigonometric and Bessel functions. This allows us to construct a monogenic pseudo‐exponential in Clifford analysis. There is a deep difference between odd and even dimensions. Copyright © 2006 John Wiley & Sons, Ltd.     

Minkowski Dimension and Cauchy Transform in Clifford Analysis

In the present paper we provide some conditions of a geometrical character for continuous extendibility of the Clifford–Cauchy transform to the boundary of a domain in the Euclidean space of higher dimensions if its density satisfies a H?lder condition. The criterion obtained in this work is an extension to a very general class of domains of a result, which has already become classical, obtained by Viorel Iftimie, who proved in 1965, for the case of a domain with compact Liapunov boundary, that the Clifford–Cauchy transform has H?lder–continuous limit values for any H?lder–continuous density. Received: August 15, 2006. Accepted: November 2, 2006.     

Boundary properties for several singular integral operators in real Clifford analysis

On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Hlder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Hlder continuous property on the boundary.     

On theT(1)-theorem for the Cauchy integral

The main goal of this paper is to present an alternative, real variable proof of theT(1)-theorem for the Cauchy integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of theT(1)-theorem. An example shows that theL ^∞-BMO estimate for the Cauchy integral does not follow fromL ² boundedness when the underlying measure is not doubling.     

On Boundary Behavior of the Cauchy Type Integrals with Values in a Universal Clifford Algebra

In this paper, we discuss boundary behavior for the Cauchy type integrals with values in a universal Clifford algebra for certain distinguished boundary and obtain some Sochocki–Plemelj formulae and Privalov–Muskhelishvili theorems.