Bilinear operators associated with Schrödinger operators

Let L = -Δ +V be a Schrödinger operator in ℝ^d and H_L¹(ℝ^d) be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T ^- defined by T^±(f,g)(x) = (T₁f)(x) (T₂g)(x) ± (T₂f)(x)(T₁g)(x), where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T ^- is bounded from L^p(ℝ) × L^q(ℝ ^d) to H_L¹(ℝ^d) for 1 < p, q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.