Heat maximal function on a Lie group of exponential growth

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.