Dixan Peña Peña

Qrama - Ghent University · Department of Mathematics

Fueter’s theorem discloses a remarkable connection existing between holomorphic functions and monogenic functions in \(\mathbb R^{m+1}\) when m is odd. It states that \(\Delta _{m+1}^{k+\frac{m-1}{2}}\bigl [\bigl (u(x_0,\vert \underline{x}\vert )+\frac{\underline{x}}{\vert \underline{x}\vert }\,v(x_0,\vert \underline{x}\vert )\bigr )P_k(\underline{...

In this paper we study two-sided (left and right) axially symmetric solutions of a generalized Cauchy–Riemann operator. We present three methods to obtain special solutions: via the Cauchy–Kowalevski extension theorem, via plane wave integrals and Funk–Hecke’s formula and via primitivation. Each of these methods is effective enough to generate all...

The Hermitian monogenic system is an overdetermined system of two Dirac type operators in several complex variables generalizing both the holomorphic system and the real Dirac system. Due to the fact that it is overdetermined, the Cauchy–Kowalevski extension problem only has a solution if the Cauchy data satisfy certain constraints. There is howeve...

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 propertie...

In this paper we study some new special functions that arise naturally within the framework of Hermitian Clifford analysis, which concerns the study of Dirac-like systems in several complex variables. In particular we focus on Hermite polynomials, Bessel functions and generalized powers. We also derive a Vekua system for solutions of Hermitian syst...

Funk-Hecke's formula allows a passage from plane waves to radially invariant functions. It may be adapted to transform axial monogenics into biaxial monogenics that are monogenic functions invariant under the product group SO(p)xSO(q). Fueter's theorem transforms holomorphic functions in the plane into axial monogenics, so that by combining both re...

Funk-Hecke's formula allows a passage from plane waves to radially invariant functions. It may be adapted to transform axial monogenics into biaxial monogenics that are monogenic functions invariant under the product group SO(p)xSO(q). Fueter's theorem transforms holomorphic functions in the plane into axial monogenics, so that by combining both re...

In this paper two important classes of orthogonal polynomials in higher dimensions using the framework of Clifford analysis are considered, namely the Clifford-Hermite and the Clifford-Gegenbauer polynomials. For both classes an explicit generating function is obtained.

In this paper we provide an alternative method to construct the Fueter primitive of an axial monogenic function of degree $k$, which is complementary to the one used in [F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter mapping theorem in integral form using spherical monogenics, Israel Journal of Mathematics, 2012]. As a byproduct, we obta...

In this paper a closed formula for the generating function of the Clifford-Gegenbauer polynomials is obtained using a combination of the Cauchy-Kowalevski extension technique and Fueter's theorem.

In this paper a new holomorphic extension theorem is presented using Clifford analysis. KeywordsClifford analysis–Isotonic functions–Holomorphic extension

The aim of this paper is to present a generalization of the Appell sequences within the framework of Clifford analysis called shifted Appell sequences. It consists of sequences {M n (x)}n ≥ 0 of monogenic polynomials satisfying the Appell condition (i.e. the hypercomplex derivative of each polynomial in the sequence equals, up to a multiplicative c...

Let Δ be the Laplace operator in ℝ 2n . The aim of this paper is to present an integral representation formula for the solutions of the generalized isotonic system ∂ ̲ 1 Δ k f+iΔ k f ˜∂ ̲ 2 =0,k∈ℕ 0 , where ∂ ̲ 1 ,∂ ̲ 2 are Dirac type operators and where f ˜ stands for the main involution in the complex Clifford algebra ℂ n . Two special cases of t...

In this paper we present the most general form of Fueter’s theorem obtained so far. Some examples of applications are also computed. Keywords.Clifford analysis-Fueter’s theorem

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions \(f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}\) of the so called isotonic system $$\partial _{\underbar{x}_1 } + i \tilde{f} \mathop{\partial _{\underbar{x} _2 } = 0}$$. The aim of this paper is to bring together these two area...

Solutions to the Dirac equation are obtained by considering functions of axial type. This indeed gives rise to Vekua systems that can be solved in terms of special functions. In this paper we investigate axial symmetry for the solutions of the two-sided monogenic system and we give examples involving Bessel functions. Comment: 8 pages, submitted fo...

This paper is devoted to a factorization of the higher dimensional Schrodinger operator in the framework of Clifford analysis.

In this paper is extended the original theorem by Fueter-Sce (assigning an $\mathbb R_{0,m}$-valued monogenic function to a $\mathbb C$-valued holomorphic function) to the higher order case. We use this result to prove Fueter's theorem with an extra monogenic factor $P_k(x_0,\underline x)$. Comment: 11 pages, accepted for publication in Journal of...

In this paper we present a closed formula for the CK-extension of the Gaussian distribution in $\mathbb R^m$, and the monogenic version of the holomorphic function $\exp(z^2/2)/z$ which is a fundamental solution of the generalized Cauchy-Riemann operator.

In this paper we prove a Cauchy-Kowalevski theorem for the functions satisfying the system DfD=0 (called inframonogenic functions). Comment: 7 pages, accepted for publication in Mathematical Journal of Okayama University

In this paper we will focus on power series expansions around special surfaces of codimension two, in particular around spheres and products of spheres. This will include a version of the Cauchy–Kowalewski extension theorem around these surfaces. This higher codimension result was only obtained so far in the flat case (see Partial Differential Equa...

We use the so-called isotonic functions to obtain extension theorems in the framework of biregular functions of Clifford analysis. In this context we also prove the Plemelj–Sokhotski formulae for the Bochner–Martinelli integral and an expression for the square of its singular version. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper we investigate a new overdetermined system in ℝm+1, called RicSom system, arising from adding one extra real dimension to the Hermitian Dirac system in ℝm, m = 2n, that uses the complex structure of ℂn. For this new system we consider a CK-extension type problem.

We consider Hölder continuous circulant (2×2) matrix functions G21 defined on the Ahlfors-David regular boundary Γ of a domain Ω in ℝ2n. The main goal is to study under which conditions such a function G21 can be decomposed as G21=G21+-G21-, where the components G21± are extendable to two-sided H-monogenic functions in the inter...

In this paper, we extend to the theory of functions of several complex variables, a theorem due to Davydov from classical complex analysis. We prove the following: if Omega subset of C-n is a bounded domain with boundary partial derivative Omega of finite (2n - 1)-dimensional Hausdorff measure H2n-1 and f is a continuous complex-valued function on...

We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in Rm, m=2n. Necessary and sufficient conditions for the solvability of this problem are obtained.

This paper the Riemann boundary value problem for solutions of special case of Beltrami equation in a bounded simply connected domain with a regular boundary in the complex plane is discussed. We obtain necessary and sufficient conditions for existence of a finite numbers of solutions of the Riemann boundary value problem. The results are applied t...

Starting with an integral representation for the class of continuously differentiable solutions \({f:{\mathbb{R}^{2n}}\, \rightarrow \, \mathbb{C}_{0,n}}\) of the system $${\partial_ {\underline{x}_{1}}f\, + \, i\tilde{f}\partial_{\underline{x}_{2}}\, = \,0}$$ where \({\mathbb{C}_{0,n}}\) is the complex Clifford algebra constructed over \({\mathbb{...

New series developments for monogenic functions are presented. The terms of these series have factors that are expressible as power functions vanishing on special higher codimension submanifolds of Euclidean space. These series are closely related with the Cauchy-Kowalewski extension problem as well as to special Vekua systems arising from the cons...

In this article the jump problem for monogenic functions (Clifford holomorphicity) on the boundary of a Jordan domain in Euclidean spaces is investigated. We shall establish some criteria that imply the uniqueness of the solution in terms of a natural analogue of removable singularities in the plane to ℝn+1 (n ≥ 2). Sufficient conditions to extend...

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.

In this paper we introduce the quaternionic Witt basis in \mathbbRm=\mathbbHÄ\mathbbR\mathbbR+m, m=4n{\mathbb{R}_m=\mathbb{H}\otimes_{\mathbb{R}}\mathbb{R}+m, m=4n} . We then define a notion of quaternionic hermitian vector derivative which leads to hermitian monogenic functions. We study the resolutions associated to quaternionic hermitian syste...

In this paper we present a generalization to Clifford analysis of Davydov’s theorem in the isotonic setting. The result obtained contains as special case the Davydov’s theorem for the Bochner-Martinelli transform.

Fueter’s Theorem on the construction of monogenic quaternionic functions starting with a holomorphic function in the upper half of the complex plane, is further generalized in a Clifford analysis setting. The result obtained contains previous generalizations as special cases.

We develop the Plemelj–Sokhotski formulas for solutions of a special case of the Beltrami equation in the classical complex analysis. These formulas relate to a principal boundary value problem for β-analytic functions, the so-called jump problem. In connection with removable singularities for the β-analytic functions, the uniqueness of the solutio...

In this article we establish an alternative proof of the generalized Fueter method presented in a former paper [Qian, T. and Sommen, F., 2003, Deriving harmonic functions in higher dimensional spaces. Zeitschrift fur Analysis und ihre Anwendungen, 22(2), 275–288] leading to the construction of special harmonic and monogenic functions in higher dime...

Integral representation formulas play an essential role in the modern function theory. They serve to solve boundary value problems for differential equations. As an example of such integral representations is the Cauchy formula for analytic functions but it, thus, is just a special case of the Cauchy–Pompieu formula. Higher Cauchy–Pompieu formulas...

The Riemann boundary value problem for solutions of a special case of the Beltrami equation in a bounded simply connected domain with a regular boundary in ℂ is discussed. The equation under consideration is properly elliptic ∂f ∂z ¯=βz z ¯∂f ∂z,z=x+iy, where 0≤β<1 (ellipticity condition). We obtain necessary and sufficient conditions for the exist...

In the recent years the so-called hyperanalytic functions theory, i.e., null solutions of the Douglis operator in , has emerged as an increasingly important area of activity for mathematicians.The definition of conjugate hyperharmonic Douglis algebra-valued functions presented in this work is shown to be a generalization of the classical conjugate...

. The main goal of this paper is centred around the study of the behavior of the Cauchy type integral and its corresponding singular version, both over nonsmooth domains in Euclidean space. This approach is based on a recently developed quaternionic Cauchy integrals theory [1, 5, 7] within the three-dimensional setting. The present work involves th...

The Riemann boundary value problem for analytic functions in the complex plane is a well established field and in this paper we are concerned with a higher dimensional version of it. The aim of this text is to give an up-to-date account on the study of the Riemann problem in the sense of Clifford analysis and domains with boundary complicated geome...

In 1.his pa¡rer the Riernann Boundarl'Value Problen on a rectifi-able open crrrve admitting the case of the non-smooth curvc is studied. The solution of thc probler¡l is constructed explicitlv under some weak restric-ti<¡ns about th¡r cocfficients of the problcnr antl using a definitiolr of index, valid for anv open rectifiabk: .Jordan curvc.