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Let L = -Δ + V be a Schrödinger operator in ${ℝ}^d$ and $H{¹}_L\left({ℝ}^d\right)$ be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by $T^{\pm }(f,g)\left(x\right)=\left(T₁f\right)\left(x\right)\left(T₂g\right)\left(x\right)\pm \left(T₂f\right)\left(x\right)\left(T₁g\right)\left(x\right)$, 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 $L^p\left({ℝ}^d\right)\times L^q\left({ℝ}^d\right)$ to $H{¹}_L\left({ℝ}^d\right)$ 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.

We first show that a linear operator which is bounded on $L{²}_w$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h{¹}_w$ if and only if this operator is uniformly bounded on all $h{¹}_w$-atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h{¹}_w$.