  Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$
endowed with the Riemannian symmetric space structure. Let $X_0, X_1, X_2$ be a
distinguished basis of left-invariant vector fields of the Lie algebra of $G$
and define the Laplacian $\Delta=-(X_0^2+X_1^2+X_2^2)$. In this paper we
consider the first order Riesz transforms $R_i=X_i\Delta^{-1/2}$ and
$S_i=\Delta^{-1/2}X_i$, for $i=0,1,2$. We prove that the operators $R_i$, but
not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show
that the second order Riesz transforms $T_{ij}=X_i\Delta^{-1}X_j$ are bounded
from $H^1$ to $L^1$, while the Riesz transforms $S_{ij}=\Delta^{-1}X_iX_j$ and
$R_{ij}=X_iX_j\Delta^{-1}$ are not.
