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

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2({\mathbb R}^n)$ with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space $H_L^1$ by means of an area integral function associated with the operator $L$. By using a variant of the maximal function associated with the semigroup $\{e^{-tL}\}_{t\geq 0}$, a space ${\rm BMO}_L$ 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 $L^2({\mathbb R}^n)$, then the dual space of $H_L^1$ is ${\rm BMO}_{L^{\ast}}$ where $L^{\ast}$ is the adjoint operator of $L$. We then obtain a characterization of the space ${\rm BMO}_L$ in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces ${\mathcal K}_L$ 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 ${\rm BMO}_L$ spaces associated with operators.