Spherical Means and Distributions in Clifford Analysis

Clifford analysis is a higher-dimensional function theory offering a refinement of classical harmonic analysis. It is centred around monogenic functions, i.e. null solutions of the rotation invariant vector-valued Dirac operator. In this context, generalizations of classical orthogonal polynomials on the real line have been introduced in order to use them as building blocks for wavelets, as has been the case, e.g. for the generalized Clifford–Hermite polynomials. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the standard Euclidean case; it focuses on simultaneous null solutions of two Hermitean Dirac operators which are invariant under the action of the unitary group. In this Hermitean setting, generalized Hermitean Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Hermitean Dirac operators. In this paper, we establish their mutual orthogonality relations w.r.t. a well-chosen weight function.

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.

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space ℝm+1 was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane. In this construction the distributional limits of these potentials at the boundary ℝm are crucial. The remarkable relationship between these distributional boundary values and four basic pseudodifferential operators linked with the Dirac and Laplace operators is studied.KeywordsFundamental SolutionLaplace OperatorDirac OperatorPseudodifferential OperatorClifford AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The monogenic functional calculus is a means of constructing functions of a finite system of bounded or unbounded operators. For bounded noncommuting operators, a polynomial function produces a polynomial of the operators in which all possible operator orderings are equally weighted. For example, for two bounded selfadjoint operators A 1, A 2, the operator \(\frac{1} {2}(A_{1}A_{2} + A_{2}A_{1})\) is associated with polynomial \(z\mapsto z_{1}z_{2}\) in two variables by the monogenic functional calculus. The same formula applies just when the spectrum \(\sigma (\xi _{1}A_{1} +\xi _{2}A_{2})\) of a finite linear combination \(\xi _{1}A_{1} +\xi _{2}A_{2}\) of A 1 and A 2 is a subset of the real numbers for any \(\xi _{1},\xi _{2} \in \mathbb{R}\). The article begins with a discussion of Clifford algebras and Clifford analysis and points out the connection with Weyl’s functional calculus for a finite system of selfadjoint operators in Hilbert space. The construction of the Cauchy kernel in the monogenic functional calculus is achieved with the plane wave decomposition of the Cauchy kernel in Clifford analysis. The formulation applies to any finite system \((A_{1},\ldots ,A_{n})\) of closed unbounded operators such that the spectrum \(\sigma \left (\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\right )\) of the closed operator \(\xi _{1}A_{1} +\ldots +\xi _{n}A_{n}\) is contained in a two-sided sector in \(\mathbb{C}\) for almost all \(\xi \in \mathbb{R}^{n}\). The connection with harmonic analysis and irregular boundary value problems is emphasized. The monogenic functional calculus may be applied to the finite commuting system \(D_{\Sigma } = i(D_{1},\ldots ,D_{n})\) of differentiating operators on a Lipschitz surface \(\Sigma \) so that an H ∞-functional calculus \(f\mapsto f(D_{\Sigma })\) is obtained for functions f uniformly bounded and left monogenic in a sector containing almost all tangent planes to \(\Sigma \).KeywordsBounded Linear OperatorClifford AlgebraFunctional CalculusSelfadjoint OperatorMonogenic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Rotations in discrete 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.

In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra $${\mathfrak {osp}}(1|2)$$ in terms of a so-called radially deformed Dirac operator $${\mathbf {D}}$$ depending on a deformation parameter c such that for $$c=0$$ the classical Dirac operator is reobtained. In this paper, several versions of the Paley–Wiener theorems for this radially deformed Fourier transform are investigated, which characterize the supports of functions associated to this generalized Fourier transform in Clifford analysis.

We develop a rigorous algebraic–analytic framework for multidual complex numbers DCn within the setting of Clifford analysis and establish a comprehensive theory of hyperholomorphic multidual complex-valued functions. Our main contributions are (i) a fully coupled multidual Cauchy–Riemann system derived from the Dirac operator, yielding precise differentiability criteria; (ii) generalized conjugation laws and the associated norms that clarify metric and geometric structure; and (iii) explicit operator and kernel constructions—including generalized Cauchy kernels and Borel–Pompeiu-type formulas—that produce new representation theorems and regularity results. We further provide matrix–exponential and functional calculus representations tailored to DCn, which unify algebraic and analytic viewpoints and facilitate computation. The theory is illustrated through a portfolio of examples (polynomials, rational maps on invertible sets, exponentials, and compositions) and a solvable multidual boundary value problem. Connections to applications are made explicit via higher-order automatic differentiation (using nilpotent infinitesimals) and links to kinematics and screw theory, highlighting how multidual analysis expands classical holomorphic paradigms to richer, nilpotent-augmented coordinate systems. Our results refine and extend prior work on dual/multidual numbers and situate multidual hyperholomorphicity within modern Clifford analysis. We close with a concise summary of notation and a set of concrete open problems to guide further development.

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2. A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. Orsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. Kahler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs.

We define a q-deformation of the Dirac operator, inspired by the one-dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace operator. This allows us to construct q-deformed Schrödinger equations in higher dimensions. The equivalence of these Schrödinger equations with those defined on q-Euclidean space in quantum variables is shown. We also define the m-dimensional q-Clifford–Hermite polynomials and show their connection with the q-Laguerre polynomials. These polynomials are orthogonal with respect to an m-dimensional q-integration, which is related to integration on q-Euclidean space. The q-Laguerre polynomials are the eigenvectors of an suq(1|1)-representation.

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.

In this paper we develop a plane system of first-order differential equations, describing nullsolutions of the Dirac operator in biaxially symmetric domains. The solutions to the equations are of pseudoanalytic type and may be used to construct on orthogonal basis for the spaces of so-called spherical monogenics. which are related to spherical harmonics with spin. Finally, after certain restrictions on dimensins. We give a complete characterization of these pscudoanalytic functions in terms of holomorphic ones. This paper may thus ge considered as a bridge between Clifford analysis and the theory of pscudoanalytic functions.

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.

Spin actions in Euclidean and Hermitian Clifford analysis in superspace