On some properties of the Hilbert transform in Euclidean space

Abstract

First some properties are recalled of the Hilbert transform introduced in the 1990's within the framework of Clifford analysis. Furthermore algebraic and geometric characterizations are given for this operator to be unitary. Special attention is paid to the Hilbert transform on the unit sphere $S^m$ and the hyperplane $\mathbb R^m$ in $\mathbb R^{m+1}$ and classical results in the plane are revisited.