Generalized Moisil-Théodoresco systems and Cauchy integral decompositions
type: Item
instance of: scholarly article
title: Generalized Moisil-Théodoresco systems and Cauchy integral decompositions
zbMATH Open document ID: 1160.30026
DOI: 10.1155/2008/746946
author: Richard Delanghe
author: Ricardo Abreu-Blaya
author: Juan Bory-Reyes
author: Frank Sommen
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/55307
review text: The authors study generalized Moisil-Théodoresco systems. Let \(\mathbb{R}^{(s)}_{0,m+1}\) be the space of \(s\)-vectors \((0\leq s\leq m+ 1)\) in the Clifford algebra \(\mathbb{R}_{0,m+1}\) constructed over the quadratic vector space \(\mathbb{R}^{0,m+1}\), let \(r,p,q\in\mathbb{N}\) with \(0\leq r\leq m+1\), \(0\leq p\leq q\) and \(r+ 2q\leq m+1\) and let \[ \mathbb{R}^{(r,p,q)}_{0,m+1}= \sum^q_{j= p}\bigoplus \mathbb{R}^{(r+ 2j)}_{0,m+1}. \] The Dirac operator \(\partial_x\) is defined by \(\partial_x= \sum^m_{i=0} e_i\partial_{x_i}\), where \(x= (x_0,x_1,\dots, x_m)\in \mathbb{R}^{m+1}\) and \(e= (e_i: i= 0,\dots, m)\) is an orthogonal basis for the quadratic space \(\mathbb{R}^{0,m+1}\). Then an \(\mathbb{R}^{(r,p,q)}_{0,m+1}\)-valued smooth function \(W\) defined in an open subset \(\Omega\subset\mathbb{R}^{m+1}\) is said to satisfy the generalized Moisil-Théodoresco system of type \((r,p,q)\) if \(\partial_x w= 0\) in \(\Omega\), where \(\partial_x\) is the Dirac operator in \(\mathbb{R}^{m+ 1}\). The authors study and characterize the structure of the solutions to Moisil-Théodoresco systems. A structure theorem is proved in Section 3 for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if \(\Omega\) is bounded with boundary \(\Gamma\), where \(\Gamma\) is an Ahlfors-David regular surface, and if \(w\) is a \(\mathbb{R}^{(r,p,q)}_{0,m+1}\)-valued Hölder continuous function on \(\Gamma\), then necessary and sufficient conditions are given under which \(w\) admits on \(\Gamma\) a Cauchy integral decomposition \(w= w_++ w_-\).
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 5313295
zbMATH Keywords: Clifford algebra
zbMATH Keywords: generalized Moisil-Théodoresco systems
Wikidata QID: Q58644828
reviewed by: A. Yu. Timofeev
MaRDI profile type: Publication
OpenAlex ID: W2061112346
cites work: Q3869803
cites work: Q4219742
cites work: On the structure of solutions of the Moisil-Théodoresco system in Euclidean space
cites work: On the solutions of the Moisil–Théodoresco system
cites work: The brothers riesz theorem for conjugate differential forms in
cites work: Duality of the space of germs of harmonic vector fields on a compact
cites work: Some examples of conjugate \(P\)-harmonic differential forms
cites work: On a annihilators of harmonic vector fields
cites work: Q4668159
cites work: On the structure of harmonic multi-vector functions
cites work: Q4478680
cites work: Q4003806
cites work: Q3995205
cites work: Function theory in plane and space
cites work: On conjugate harmonic functions in Euclidean space
cites work: Q5487699
cites work: Q5493586
cites work: Q5549261
cites work: Die Funktionentheorie der Differentialgleichungen \(\Delta u=0\) und \(\Delta\Delta u=0\) mit vier reellen Variablen
cites work: Quaternionic analysis
cites work: Q3844606
cites work: Q5565773
cites work: Q4328334
cites work: Q4278678
cites work: Commutators and singular integral operators in Clifford analysis
cites work: On the Riemann Hilbert type problems in Clifford analysis
cites work: Clifford Cauchy type integrals on Ahlfors-David regular surfaces in \(\mathbb R^{m+1}\)
cites work: Harmonic multivector fields and the Cauchy integral decomposition in Clifford analysis
cites work: Jump problem and removable singularities for monogenic functions
cites work: Q3310867
cites work: Q5567281
cites work: The Clifford–Cauchy transform with a continuous density: N. Davydov's theorem
cites work: Q3509307
cites work: Minkowski dimension and Cauchy transform in Clifford analysis
cites work: Cauchy transform on nonrectifiable surfaces in Clifford analysis
cites work: Weighted Cauchy transforms in Clifford analysis
cites work: Cauchy transform and rectifiability in Clifford analysis
cites work: Cauchy integral decomposition of multi-vector valued functions on hypersurfaces
description: scientific article; zbMATH DE number 5313295