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

We investigate the Hardy space HL1 associated with the Schrödinger operator L=−Δ+VL=−Δ+V on RnRn, where V=∑j=1dVj. We assume that each VjVj depends on variables from a linear subspace VjVj of RnRn, dimVj≥3dimVj≥3, and VjVj belongs to Lq(Vj)Lq(Vj) for certain qq. We prove that there exist two distinct isomorphisms of HL1 with the classical Hardy space. We deduce as a corollary a specific atomic characterization of HL1. We also prove that the space HL1 can be described by means of the Riesz transforms RL,i=∂iL−1/2RL,i=∂iL−1/2.