On the Hilbert transform and conjugate harmonic functions
by
Fred Brackx
Ghent University, Clifford Research Group
Coauthors: Hennie De Schepper, David Eelbode

On the real line the Hilbert transform H: L₂(R) ® L₂(R), given by

H[f](x) = 1 p  Pv ó õ +¥ -¥  f(t) x-t  dt

and the Poisson transform P: L₂(R) ® Harm²(C⁺), given by

P[f](x) = P(·, y) *f(·)(x) = 1 p   ó õ +¥ -¥  y (x-t)2 + y2  f(t)  dt

are related by the well-known property that, for real-valued functions f, P[f] and P[H[f]] are conjugate harmonic functions in the upper half plane C⁺, and constitute the real and imaginary parts of the holomorphic Cauchy integral C : L₂(R) ® H²(C⁺), given by

C[f](z) = - 1 2pi ó õ + ¥ - ¥  f(t) (x-t) + iy  dt = 1 2 P[f] + i 2 P [ H [f]]

If W is a bounded, simply connected domain in the complex plane, with C^¥ smooth boundary, then the analogue of this property is precisely used to define the Hilbert transform on ¶W. Indeed, take u Î C^¥(¶W) real-valued, then there exists a real-valued harmonic function U Î C^¥([`(W)] ) for which the restriction to the boundary ¶W is the given function u. Let V Î C<sup>¥</sup>([`(W)]) be the conjugate harmonic to U and let v be the restriction to the boundary ¶W of V, then v is called the Hilbert transform of u. This Hilbert transform maps C^¥(¶W) into itself and extends uniquely to a bounded linear operator on L₂(¶W).

In this contribution, we investigate how the above relationship between the Hilbert transform and the notion of conjugate harmonic functions behaves in higher dimension within the framework of Clifford analysis. We first treat the special cases of the half space and the unit ball, passing then to a more general bounded domain in R^m+1, with C^¥ smooth boundary.

Date received: March 31, 2005