Hardy spaces and Riesz transforms on a Lie group of exponential growth

Let $G$ be the Lie group ${\Bbb{R}}^2\rtimes {\Bbb{R}}^+$ endowed with the Riemannian symmetric space structure. Take a distinguished basis $X_0,\, X_1,\,X_2$ of left-invariant vector fields of the Lie algebra of $G$, and consider the Laplacian $\Delta=-\sum_{i=0}^2X_i^2$ and the first-order Riesz transforms $\mathcal R_i=X_i\Delta^{-1/2}$, \hskip3pt $i=0,1,2$. We first show that the atomic Hardy space $H^1$ in $G$ introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms $\mathcal R_i$. It is also proved that two of these Riesz transforms are bounded from $H^1$ to $H^1$.