Regular functions of several quaternionic variables and the Cauchy-Fueter complex
type: Item
instance of: scholarly article
title: Regular functions of several quaternionic variables and the Cauchy-Fueter complex
zbMATH Open document ID: 0966.35088
DOI: 10.1007/BF02923085
published in: The Journal of Geometric Analysis
publication date: 2000-12-11T00:00:00Z
review text: This remarkable paper contains in his first part a new proof of the generalization of the famous Hartogs theorem in the \(n\)-dimensional quaternionic space \(\mathbb{H}^n\) \((n> 1)\), i.e. a regular function on \(\mathbb{H}^n\setminus K\) (\(K\) compact convex subset of \(\mathbb{H}^n\)) can be uniquely regular extended on \(\mathbb{H}^n\). This proof is based on an idea of L. Ehrenpreis. Because of the non-commutativity of the ``partial'' quaternionic derivatives \[ {\partial\over\partial\overline q_1},\dots, {\partial\over\partial\overline q_n}\quad (q_1,\dots, q_n)\in \mathbb{H}^n, \] it is necessary to use matrix techniques. One has to check that the cokernel of a corresponding symbol matrix is torsion-free which again can be obtained when its minors are relatively prime. Furthermore, for the Cauchy-Fueter system \[ {\partial f_i\over\partial\overline q_i}= g_i\quad (i= 1,2) \] are got (very impressive) necessary and sufficient solvability conditions. At the end new duality theorems are proved and similarities to Sato's classical results are lined out.
Mathematics Subject Classification ID: 35N15
Mathematics Subject Classification ID: 30G99
Mathematics Subject Classification ID: 58J10
zbMATH DE Number: 1542090
zbMATH Keywords: Hartogs theorem
zbMATH Keywords: matrix techniques
zbMATH Keywords: new duality theorems
author: William W. Adams
author: Carlos A. Berenstein
author: Philippe Loustaunau
author: Irene Sabadini
author: D. C. Struppa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1542090