On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis

Euclidean Clifford analysis is a higher dimensional function theory, refining harmonic analysis, centred around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator, called the Dirac operator. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions of two Hermitean Dirac operators, invariant under the action of the unitary group. In this paper, a Cauchy integral formula is established by means of a matrix approach, allowing the recovering of the traditional Martinelli-Bochner formula for holomorphic functions of several complex variables as a special case.     

On Fundamental Solutions in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ?{\underline{\partial}} called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ?_z{\partial_{\underline{z}}} and ?_z^f{\partial_{\underline{z}^\dagger}} which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford–Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ?E = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.     

Generalized Hermitean Clifford–Hermite polynomials and the associated wavelet transform

Clifford analysis is a higher‐dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing higher‐dimensional continuous wavelet transforms, the construction of the wavelets being based on generalizations to a higher dimension of classical orthogonal polynomials on the real line. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focusses on so‐called Hermitean monogenic functions, i.e. simultaneous null solutions of two Hermitean Dirac operators. In this Hermitean setting, Clifford–Hermite polynomials and their associated families of wavelet kernels have been constructed starting from a Rodrigues formula involving both Hermitean Dirac operators mentioned. Unfortunately, the property of the so‐called vanishing moments of the corresponding mother wavelets, ensuring that polynomial behaviour in the analyzed signal is filtered out, is only partially satisfied and has to be interpreted with care, the underlying mathematical reason being the fact that the Hermitean Clifford–Hermite polynomials show a too restrictive structure. In this paper, we will remediate this drawback by considering generalized Hermitean Clifford–Hermite polynomials, involving in their definition homogeneous Hermitean monogenic polynomials. The ultimate goal being the construction of new continuous wavelet transforms by means of these polynomials, we first deeply investigate their properties, amongst which are their connection with the traditional Laguerre polynomials, their structure and recurrence relations. Copyright © 2008 John Wiley & Sons, Ltd.     

A Hermitian Cauchy formula on a domain with fractal boundary

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called Dirac operator, which factorizes the Laplacian; monogenic functions may thus also be seen as a generalization of holomorphic functions in the complex plane. Hermitian Clifford analysis offers yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitian (or h-) monogenic functions, of two Hermitian Dirac operators which are invariant under the action of the unitary group. In Brackx et al. (2009) [8] a Clifford-Cauchy integral representation formula for h-monogenic functions has been established in the case of domains with smooth boundary, however the approach followed cannot be extended to the case where the boundary of the considered domain is fractal. At present, we investigate an alternative approach which will enable us to define in this case a Hermitian Cauchy integral over a fractal closed surface, leading to several types of integral representation formulae, including the Cauchy and Borel-Pompeiu representations.     

Paracomplex Hermitean Clifford Analysis

Substituting the complex structure by the paracomplex structure plays an important role in para-geometry and para-analysis. In this article we shall introduce the paracomplex structure into the realm of Clifford analysis and establish paracomplex Hermitean Clifford analysis by constructing a paracomplex Hermitean Dirac operator \({\mathcal {D}}\) and establishing the corresponding Cauchy integral formula. The theory of paracomplex Hermitean Clifford analysis turns out to be similar to that of complex Hermitean Clifford analysis which recently emerged as a refinement of the theory of several complex variables. It deserves to be pointed out that the introduction of a single operator \({\mathcal {D}}\) in the paracomplex setting has an advantage over the complex setting where complex Hermitean monogenic functions are described by a system of equations instead of being given as null-solution of a single Dirac operator as in the case of classic monogenic functions.     

The Hermitean Hilbert–Dirac Connection

Hermitean Clifford analysis is a recent branch of Clifford analysis, refining the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean monogenic functions, of two complex Dirac operators which are invariant under the action of the unitary group. The specificity of the framework, introduced by means of a complex structure creating a Hermitean space, forces the underlying vector space to be even dimensional. Thus, any Hilbert convolution kernel in should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in . In this paper we show that the difficulties posed by this inevitable dimensional jump can be overcome by following a matrix approach. The resulting matrix Hermitean Hilbert transform also gives rise, through composition with the matrix Dirac operator, to a Hermitean Hilbert–Dirac convolution operator “factorizing” the Laplacian and being closely related to Riesz potentials. Received: October, 2007. Accepted: February, 2008.     

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the framework of circulant (2×2) matrix functions. Under this setting we will present the half Dirichlet problem for circulant (2×2) matrix functions on the unit ball of even dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solution to the Dirichlet problem on the unit ball for circulant (2×2) matrix functions and the solution to the classical Dirichlet problem as the special case, derive a decomposition of the Poisson kernel for matrix Laplace operator, and further obtain the decomposition theorems of solution space to the Dirichlet problem for circulant (2×2) matrix functions.     

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.     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

The Cauchy-Kovalevskaya extension theorem in Hermitian Clifford analysis

Hermitian Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitian monogenic functions, of two Hermitian conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitian monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitian monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitian monogenics, i.e. homogeneous Hermitian monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.     

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.     

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.     

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.     

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 invariant differential operator (θ) ca...     

Duality for Hermitean Systems in {\mathbb R^{2n}}

In this paper, using the ring structure of the space of circulant (2 × 2)-matrix, we characterize the dual of the (Fréchet) space of germs of left Hermitean monogenic matrix functions in a compact set _boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose_boxclose R^2n{{\bf E}\subset\mathbb R^{2n}}. As an application we describe the dual space of the so-called h-monogenic functions satisfying simultaneously two Dirac type equations.     

A Hilbert Transform for Matrix Functions on Fractal Domains

We consider H?lder continuous circulant (2 × 2) matrix functions G¹₂{{\bf G}^1_2} defined on the fractal boundary Γ of a Jordan domain Ω in \mathbbR²ⁿ{\mathbb{R}^{2n}}. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function G¹₂{{\bf G}^1_2} in the interior and exterior of Ω, in terms of its boundary value g¹₂=G¹₂|_G{{\bf g}^1_2={\bf G}^1_2|_\Gamma}, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular.     

Hypercomplex Functional Calculi and Function Theories in Clifford Analysis

Some properties of hypercomplex functions (the null solutions of the polynomial Dirac operators in Rⁿ⁺¹) in Clifford Analysis are discussed, their hypercomplex functional calculi for an n-tuple non-commuting self-adjoint operators A are constructed by the use of Cauchy integral formulas, the polynomial approaches to functional calculi are also considered. Although these hypercomplex function theories have different representative forms, their hypercomplex functional calculi are the same as the monogenic functional calculus.     

Cauchy-Kowalevski extensions and monogenic plane waves using spherical monogenics

Clifford analysis may be regarded as a direct and elegant generalization to higher dimensions of the theory of holomorphic functions in the complex plane, centred around the notion of monogenic function, i.e. a null solution of the Dirac operator. This paper dealswith axial and biaxialmonogenic functions containing sphericalmonogenics. They are constructed bymeans of two fundamentalmethods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves.     

Regular functions of several quaternionic variables and the Cauchy-Fueter complex

We employ a classical idea of Ehrenpreis, together with a new algebraic result, to give a new proof that regular functions of several quaternionic variables cannot have compact singularities. As a byproduct we characterize those inhomogeneous Cauchy-Fueter systems which admit solutions on convex sets. Our method readily extends to the case of monogenic functions on Clifford Algebras. We finally study a free resolution of the Cauchy-Fueter complex of differential operators and we obtain some new duality theorems which hint at a hyperfunction theory of several quaternionic variables.     

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.