Boundedness from H^1 to L^1 of Riesz transforms on a Lie group of exponential growth

Let G be the Lie group R^2 x R^+ endowed with the Riemannian symmetric space structure. Let X_0, X_1, X_2 be a distinguished basis of left-invariant vectori fields of the L:ie algebra of G and L be the associated Laplacian. We consider the Riesz transform of the firts order associated with L and their adjoint operators and investigate their boundedness properties on a suitable Hardy space H^1. We also study the boundedness of the second order Riesz transforms associated with L.