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

Let G be the Lie group R^2\rtimes 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 R_i=X_i\Delta^{-1/2}, 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 R_i. It is also proved that two of these Riesz transforms are bounded from H^1 to H^1.