Cauchy integral decomposition of multi-vector valued functions on hypersurfaces
type: Item
instance of: scholarly article
title: Cauchy integral decomposition of multi-vector valued functions on hypersurfaces
zbMATH Open document ID: 1089.30047
DOI: 10.1007/BF03321089
author: Richard Delanghe
author: Ricardo Abreu-Blaya
author: Juan Bory-Reyes
author: Frank Sommen
published in: Computational Methods and Function Theory
publication date: 2006-03-10T00:00:00Z
review text: Let \(\Gamma\) be a Jordan curve with certain smoothness and let \(f\) be a Hölder function on \(\Gamma\), then one of the Sokhotski--Plemelj formulas can be interpreted as a decomposition of \(f\) into the sum of two functions which are holomorphically extendable inside and outside \(\Gamma\). Trying to generalize such a decomposition to higher dimensions is closely related with the problem of an adequate multidimensional analog of holomorphic function theory in the plane. By obvious reason, complex analysis in \(\mathbb C^m\), \(m>1\), does not serve for that purpose. \textit{E. Dyn'kin} in [Complex Variables, Theory Appl. 31, No.2, 165--176 (1996; Zbl 0865.30056)], [J. Anal. Math. 73, 165--186 (1997; Zbl 0899.58001)] worked out the decomposition of a continuous \(k\)--form \(\eta_k\) on a \(C_1\)--smooth hypersurface \(\sum\) in \(\mathbb R^m\) as a sum of continuous \(k\)-- forms \(\eta_k^{\pm}\) on \(\sum\): \[ \eta_k = \eta_k^++\eta_k^- \tag{Juan} \] where (i) \(\sum\) is the boundary of an open domain \(G\) in \(\mathbb R^m\); (ii) \(\eta_k^{\pm}\) are extendable to harmonic \(k\)--forms of class \(H^s\), \(s>0\), in respectively \(G_+:=G\) and \(G_-:=\mathbb R^m\setminus\overline G\) with \(\eta_k^-(\infty)=0\). Dyn'kin gave necessary and sufficient conditions under which \(\eta_k\) can be represented by (ref {Juan}), his so--called condition (A) and condition (B). As was pointed out in [\textit{E. Dyn'kin}, J. Anal. Math. 73, 165--186 (1997; Zbl 0899.58001)], for an open domain in \(\mathbb R^m\) and \(0<s<1\), the class \(H^s\) of harmonic \(k\)-forms appearing in (B) consists of all harmonic \(k\)-forms in that domain satisfying a Hölder condition of order \(s\) in the closure of the domain. One of the main aims of the paper under review is to show that Condition (A) for harmonic \(k\)-forms is equivalent to a so-called conservation law (CL) for harmonic \(k\)-multi-vector fields. The latter is being established in Section 4, after Section 2 where some pr…
Mathematics Subject Classification ID: 30G35
zbMATH DE Number: 5011339
zbMATH Keywords: Clifford analysis
zbMATH Keywords: Cauchy transform
zbMATH Keywords: Hilbert transform
reviewed by: M. Shapiro
MaRDI profile type: Publication
cites work: Harmonic multivector fields and the Cauchy integral decomposition in Clifford analysis
cites work: Q4003806
cites work: Cauchy integral decomposition for harmonic vector fields
cites work: Cauchy integral decomposition for harmonic forms
cites work: Q3995205
cites work: Q4878738
cites work: Q4878730
cites work: Q3287742
cites work: On the theory of harmonic functions of several variables
full work available at URL: https://doi.org/10.1007/bf03321089
OpenAlex ID: W1974301494
description: scientific article; zbMATH DE number 5011339