For alpha>0 we consider the system l_k^{ (alpha-1)/2}(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf =-\frac{ d^2}{ dx^2}f-\frac{ alpha}{ x}\frac{ d}{ dx}f+x^2 f. We define an atomic Hardy space H^1_{ at}(X), which is a subspace of L^1((0,infty), x^alpha dx). Then we prove that the space H^1_{ at}(X) is also characterized by the Riesz transform Rf=\sqrt{ \pi}\frac{ d}{ dx}L^{ -1/2}f in the sense that f\in H^1_{ at}(X) if and only if f,Rf \in L^1((0,infty),x^alpha dx).
Categories: math.FA