Hardy spaces related to Schrِdinger operators with potentials which are sums of -functions

We investigate the Hardy space H L 1 associated with the Schrödinger operator L = − Δ + V on R n , where V = ∑ j = 1 d V j . We assume that each V j depends on variables from a linear subspace V j of R n , dim V j ≥ 3 , and V j belongs to L q ( V j ) for certain q . We prove that there exist two distinct isomorphisms of H L 1 with the classical Hardy space. We deduce as a corollary a specific atomic characterization of H L 1 . We also prove that the space H L 1 can be described by means of the Riesz transforms R L , i = ∂ i L − 1 / 2 .