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

Abstract

Let L L be the infinitesimal generator of an analytic semigroup on L 2 ( R n ) L^2({\mathbb R}^n) with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space H L 1 H_L^1 by means of an area integral function associated with the operator L L . By using a variant of the maximal function associated with the semigroup { e − t L } t ≥ 0 \{e^{-tL}\}_{t\geq 0} , a space BMO L \textrm {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 L has a bounded holomorphic functional calculus on L 2 ( R n ) L^2({\mathbb R}^n) , then the dual space of H L 1 H_L^1 is BMO L ∗ \textrm {BMO}_{L^{\ast }} where L ∗ L^{\ast } is the adjoint operator of L L . We then obtain a characterization of the space BMO L \textrm {BMO}_L in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces K L {\mathcal K}_L of BMO L _{ L} when L L is a second-order elliptic operator of divergence form and when L L is a Schrödinger operator, and study the inclusion between the classical BMO space and BMO L \textrm {BMO}_L spaces associated with operators.