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

Duong, Xuan¹ ; Yan, Lixin²

¹ Department of Mathematics, Macquarie University, NSW 2109, Australia
² Department of Mathematics, Macquarie University, NSW 2109, Australia and Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China

Journal of the American Mathematical Society, Tome 18 (2005) no. 4, pp. 943-973

Résumé

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 $\textrm {BMO}_L$ 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 $L$ has a bounded holomorphic functional calculus on $L^2({\mathbb R}^n)$, then the dual space of $H_L^1$ is $\textrm {BMO}_{L^{\ast }}$ where $L^{\ast }$ is the adjoint operator of $L$. We then obtain a characterization of the space $\textrm {BMO}_L$ 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 $\textrm {BMO}_L$ spaces associated with operators.

Détail

DOI : 10.1090/S0894-0347-05-00496-0

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