Riesz transform characterization of H1 spaces associated with certain Laguerre expansions

For α>0 we consider the system ψk(α−1)/2(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf=−d2dx2f−αxddxf+x2f. We define an atomic Hardy space Hat1(X), which is a subspace of L1((0,∞),xαdx). Then we prove that the space Hat1(X) is also characterized by the Riesz transform Rf=π∂∂xL−1/2f in the sense that f∈Hat1(X) if and only if f,Rf∈L1((0,∞),xαdx).