International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 98145, 19 pages

Generalized multidimensional Hilbert transforms in Clifford analysis

Fred Brackx, Bram De Knock, and Hennie De Schepper

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 Fred Brackx et al. 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.


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.