Computational Methods and Function Theory 5 (2005), No. 1, 111--134
Copyright Heldermann Verlag 2005
Ricardo Abreu Blaya
rabreu@facinf.uho.edu.cu , Faculty of Mathematics and Informatics, University of Holguín, Holguín 80100, Cuba.
Juan Bory Reyes
jbory@rect.uo.edu.cu , Department of Mathematics, University of Oriente, Santiago de Cuba 90500, Cuba.
Richard Delanghe
Richard.Delanghe@UGent.be , Department of Mathematical Analysis, University of Ghent, Galglaan 2, B-9000 Gent, Belgium.
Frank Sommen
fs@cage.ugent.be , Department of Mathematical Analysis, University of Ghent, Galglaan 2, B-9000 Gent, Belgium.
Let $\Omega$ be a bounded open and connected subset of $\mathbb{R}^m$ which has a $C_\infty$-boundary $\Sigma$ and let $F_k\in C_\infty(\Sigma)$ be a $k$-multi-vector valued function on~$\Sigma$. Under which conditions can $F_k$ be decomposed as $F_k=F_k^++F_k^-$ where $F_k^\pm$ are extendable to harmonic $k$-multi-vector fields in $\Omega_\pm$ with $\Omega_+=\Omega$ and ${\Omega_-=\mathbb{R}^m\setminus\overline\Omega}$? This question is answered by proving a set of equivalent assertions, including a conservation law on $F_k$ and conditions on the Cauchy transform $\mathcal{C}_\Sigma F_k$ and on the Hilbert transform $H_\Sigma F_k$ of $F_k$.
Keywords: Clifford analysis, multi-vector valued functions, Cauchy transform, Hilbert transform.
MSC 2000: 30G35, 45B20.
[FullText-pdf (354 K)] [FullText-ps (588 K)]