VMO spaces associated with operators with Gaussian upper bounds on product domains

In this paper, we extend the well-known result "the predual of Hardy space H¹ is VMO" to the product setting, associated with differential operators. Let L_i, i = 1, 2, be the infinitesimal generators of the analytic semigroups {e^-tLi} on L²(ℝ).  Assume that the kernels of the semigroups {e^-tLi} satisfy the Gaussian upper bounds.  We introduce the VMO spaces VMO_L1,L2(ℝ × ℝ) associated with operators L₁ and L₂ on the product domain ℝ × ℝ, then show that the dual space of VMO_L1,L2(ℝ × ℝ) is the Hardy space H¹_L1^∗_,L2^∗ (ℝ × ℝ) associated with the adjoint operators L₁^∗ and L₂^∗.