Atomic and maximal Hardy spaces on a Lie group of exponential growth

The main part of this paper is a shorter version of a joint work with P. Sj\"ogren. 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). We recall the definition and the main properties of the atomic Hardy space H^1_{at} on the group G. Then we introduce a maximal Hardy space H^1_{max,h} on G defined in terms of the maximal function associated with the heat kernel of the Laplacian \Delta. We show that the atomic Hardy space is strictly included in the heat maximal Hardy space. In the last part of the paper, which is new, we consider the maximal Hardy space H^1_{max,p} defined in terms of the Poisson kernel of the Laplacian \Delta and show that it strictly contains the atomic Hardy space H^1_{at}.