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 vector fields of the Lie algebra of G and define the Laplacian Delta = - (X-0(2) +X-1(2) +X-2(2)). In this paper we consider the first order Riesz transforms R-i = X-i Delta(-1/2) and S-i = Delta X--1/2(i) for i = 0, 1, 2. We prove that the operators Ri, but not the Si, are bounded from the Hardy space H-1 to L-1. We also show that the second-order Riesz transforms T-ij = Xi Delta(-1) Xj are bounded from H-1 to L1, while the transforms S-ij = Delta-1XiXj and R-ij = XiXj Delta(-1), for i, j = 0, 1, 2, are not