Orthogonality of the generalized Hermitean Clifford–Hermite polynomials

Hermitean Clifford analysis focusses on h–monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J ∈ SO(2n;\({\mathbb{R}}\)), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 ± iJ) project the initial basis (eα, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of Open image in new window into two components, where the SO(2n;\({\mathbb{R}}\))-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n;\({\mathbb{R}}\)), denoted Spin J (2n;\({\mathbb{R}}\)), being isomorphic with the unitary group U(n;\({\mathbb{C}}\)). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin J (2n;\({\mathbb{R}}\)). The eventual goal is to extend the complex structure J to the whole Clifford algebra Open image in new window , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.

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 \({\mathbb{R}}^{2n}\) should originate from the non-tangential boundary limits of a corresponding Cauchy kernel in \({\mathbb{R}}^{2n+2}\). 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.

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

Spin actions in Euclidean and Hermitian Clifford analysis in superspace

We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} } \) and \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }^\dag \) and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.

We consider Holder continuous circulant (2 × 2) matrix functions \({{\bf G}^1_2}\) defined on the fractal boundary Γ of a Jordan domain Ω in \({\mathbb{R}^{2n}}\). 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 \({{\bf G}^1_2}\) in the interior and exterior of Ω, in terms of its boundary value \({{\bf g}^1_2={\bf G}^1_2|_\Gamma}\), 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.

This paper consists of two parts. In the first part we deal with specific higher dimensional distributions within the framework of Clifford analysis. These distributions are "classical" in the sense that they were already introduced, albeit dispersed, in the literature on harmonic analysis and on Clifford analysis. Amongst these classical distributions are the fundamental solutions of the natural powers of the Laplace and the Dirac operators, and the integral kernel of the Hilbert transform. The strength of our approach is its unifying character.In the second part new higher dimensional distributions are introduced, generalizing the distributions of part I. Crucial to this generalization is the use of so-called surface spherical monogenics. The whole picture thus obtained offers structural clarity and unity.

In this paper, we first define Teodorescu operator related to the -Dirac operator in Clifford analysis. Then we discuss the existence, q-times integrability, differentiability and mapping properties of Teodorescu operator in a nonempty open set , which is bounded connected.

Special functions in Clifford analysis and axial symmetry

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes' theorem, Cauchy's theorem and a Cauchy integral formula are established.

Rotations in discrete Clifford analysis

The aim of the paper is to discuss a role of symmetries for equations studied in Clifford analysis and to show that important information about properties of solutions can be obtained by a study of corresponding transformation groups, their action and by using very efficient tools developed in representation theory of simple Lie groups.

Let Cℓ n be the (universal) Clifford algebra generated by e1,…,e n The Cauchy-Riemann (or Dirac) operator in Cℓ n is defined by $$ D = \sum\nolimits_{{i = 0}}^{n} {{{e}_{i}}} \frac{\partial }{{\partial {{x}_{i}}}}. $$ The functions $$ f{\text{ }}:{\text{ }}\Omega \to C{{\ell }_{{n,}}}{\text{ }}\Omega {\text{ }} \subset {\text{ }}{{\mathbb{R}}^{{n + 1}}} $$ open, satisfying Df = 0, called left monogenic functions, are widely studied in Clifford analysis. The power function xm is not monogenic. The second author noticed that the power function is the solution of the modified Cauchy-Riemann system $$ xnDf + \left( {n - 1} \right)fn = 0 $$ which has strong connections to the hyperbolic metric. We study solutions of the equation $$ xnDf + \left( {n - 1} \right)Q_{{n - 1}}^{'}\left( f \right) = 0, $$ where Qf is the function given by the decomposition $$ f\left( x \right) = Pf\left( x \right) + Qf\left( x \right){{e}_{n}} $$ for $$ Pf\left( x \right),Qf\left( x \right) \notin C{{\ell }_{{n - 1}}}. $$ If $$ f = \sum\nolimits_{{i = 0}}^{n} {{{f}_{i}}} {{e}_{i}} $$ for some real functions f i , then f is the solution of the modified Cauchy-Riemann system stated above.

Generalized Taylor series in hermitian Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: urn:x-wiley:dummy:mana201100302:equation:mana201100302-math-0001where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.