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International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 98145, 19 pages
http://dx.doi.org/10.1155/IJMMS/2006/98145

Generalized multidimensional Hilbert transforms in Clifford analysis

Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Gent 9000, Belgium

Received 26 September 2005; Accepted 29 December 2005

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though in each of these generalizations some traditional properties of the Hilbert transform are inevitably lost, new bounded singular operators emerge on Hilbert or Sobolev spaces of L2-functions.