Let L be the infinitesimal generator of an analytic semigroup on L2 (ℝ) with suitable upper bounds on its heat kernels, and L has a bounded holomorphic functional calculus on L2 (ℝ). In this article, we introduce new function spaces H L 1 (ℝ × ℝ) and BMOL(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 BMOL (ℝ × ℝ) in terms of Carleson conditions. We also show that the John-Nirenberg inequality holds for the space BMOL (ℝ × ℝ). Applications include large classes of differential operators such as the magnetic Schrodinger operators and second-order elliptic operators of divergence form or nondivergence form in one dimension.