Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains

Let L be the infinitesimal generator of an analytic semigroup on L ² (ℝ) with suitable upper bounds on its heat kernels, and L has a bounded holomorphic functional calculus on L ² (ℝ). In this article, we introduce new function spaces H _L ¹ (ℝ × ℝ) and BMO _L(ℝ × ℝ) (dual to the space H _L* ¹ (ℝ × ℝ) in which L*is the adjoint operator of L) associated with L, and they generalize the classical Hardy and BMO spaces on product domains. We obtain a molecular decomposition of function for H _L ¹ (ℝ × ℝ) by using the theory of tent spaces and establish a characterization of BMO _L (ℝ × ℝ) in terms of Carleson conditions. We also show that the John-Nirenberg inequality holds for the space BMO _L (ℝ × ℝ). Applications include large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form or nondivergence form in one dimension.