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

Let L be the infinitesimal generator of an analytic semigroup on L²(ℝⁿ) with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space H¹L by means of an area integral function associated with the operator L. By using a variant of the maximal function associated with the semigroup [equation omitted for formatting reasons], a space BMO L of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if L has a bounded holomorphic functional calculus on L²(ℝⁿ), then the dual space of H¹L is BMO L* where L* is the adjoint operator of L. We then obtain a characterization of the space BMO L in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces KL of BMO L when L is a second-order elliptic operator of divergence form and when L is a Schrödinger operator, and study the inclusion between the classical BMO space and BMO L spaces associated with operators.31 page(s