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 Δ=-∑ᵢ₌₀²Xᵢ² and the first-order Riesz transforms Rᵢ=XᵢΔ-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.