Hilbert Transformation and $r\mathrm{Spin}(n)+\mathbb{R}^n$ Group
  In this paper we study symmetry properties of the Hilbert transformation of
several real variables in the Clifford algebra setting. In order to describe
the symmetry properties we introduce the group $r\mathrm{Spin}(n)+\mathbb{R}^n,
r>0,$ 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 $r\mathrm{Spin}(n)+\mathbb{R}^n.$ In the present paper, for $n=2$
and $3$ we obtain, explicitly, the induced spinor representations of the
$r\mathrm{Spin}(n)+\mathbb{R}^n$ group. Then we decompose the natural
representation of $r\mathrm{Spin}(n)+\mathbb{R}^n$ into the direct sum of some
two irreducible spinor representations, by which we characterize the Hilbert
transformation in $\mathbb{R}^3$ and $\mathbb{R}^2.$ Precisely, we show that a
nontrivial skew operator is the Hilbert transformation if and only if it is
invariant under the action of the $r\mathrm{Spin}(n)+\mathbb{R}^n, n=2,3,$
group.
