Riesz transform characterization of H1 spaces associated with certain Laguerre expansions

For alpha > 0 we consider the system psi((alpha-1)/2)(k)(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf = -d(2)/dx(2) f - alpha/x d/dx f + x(2)f. We define an atomic Hardy space H-at(1) (X), which is subspace of L-1((0, infinity), x(alpha) dx). Then we prove that the space H-at(1) (X) is also characterized by the Riesz transform Rf = root pi partial derivative/partial derivative x L-1/2 f in the sense that f is an element of H-at(1)/ (X) if and only if f, Rf is an element of L-1 ((0, infinity), x(alpha) dx). (C) 2011 Elsevier Inc. All rights reserved.