The Symmetry of Hilbert Transformation in \(\mathbb {R}^3\)

In this paper we study symmetry properties of the Hilbert transformation of the three real variables in the quaternion setting. In order to describe the symmetry properties we introduce the group \(r\textrm{Spin}(3)+\mathbb {R}^3\) which is essentially an extension of the ax+b group. The study concludes that the Hilbert transformation has certain characteristic symmetry properties in terms of \(r\textrm{Spin}(3)+\mathbb {R}^3.\) We first obtain the spinor representation of the group induced by one of \(\textrm{Spin}(2)\) in \(\mathbb {H}\). Then we decompose the natural representation of \(r\textrm{Spin}(3)+\mathbb {R}^3\) into the direct sum of some two irreducible spinor representations, by which we characterize the Hilbert transformation in \(\mathbb {R}^3\). Precisely, we show that a nontrivial operator is essentially the Hilbert transformation if and only if it is invariant under the action of the \(r\textrm{Spin}(3)+\mathbb {R}^3\) group.