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Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$endowed with the Riemannian symmetric space structure. Let $X_0, X_1, X_2$ be adistinguished 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 weconsider the first order Riesz transforms $R_i=X_i\Delta^{-1/2}$ and$S_i=\Delta^{-1/2}X_i$, for $i=0,1,2$. We prove that the operators $R_i$, butnot the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also showthat the second order Riesz transforms $T_{ij}=X_i\Delta^{-1}X_j$ are boundedfrom $H^1$ to $L^1$, while the Riesz transforms $S_{ij}=\Delta^{-1}X_iX_j$ and$R_{ij}=X_iX_j\Delta^{-1}$ are not.

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