  Let G be the Lie group R^2\rtimes 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 show that the maximal function
associated with the heat kernel of the Laplacian \Delta is bounded from the
Hardy space H^1 to L^1. We also prove that the heat maximal function does not
provide a maximal characterization of the Hardy space H^1.
