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

Let $G$ be the Lie group ${\mathbb{R}}^2⋊{\mathbb{R}}^{+}$ 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 $\mathrm{\Delta }=-{\sum }_{i=0}^2{X}_i^2$ and the first-order Riesz transforms ${\mathcal{R}}_i=X_i{\mathrm{\Delta }}^{-1\mathrm{/}2}$, \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 $H^1$.