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

Let G be the Lie group R² R^+ endowed with the Riemannian symmetric space structure. Take a distinguished basis X₀, \, X₁, \, X₂ of left-invariant vector fields of the Lie algebra of G, and consider the Laplacian =-₈=₀²Xᵢ² and the first-order Riesz transforms Rᵢ=Xᵢ^-1/2, 3pt i=0, 1, 2. We first show that the atomic Hardy space H¹ in G introduced by the authors in a previous paper does not admit a characterization in terms of the Riesz transforms Rᵢ. It is also proved that two of these Riesz transforms are bounded from H¹ to H¹.