Tohoku Mathematical Journal 2005 June SECOND SERIES VOL. 57, NO. 2

 Tohoku Math. J. 57 (2005), 147-170

Title HARDY SPACES ASSOCIATED TO THE SECTIONS

Author Yong Ding and Chin-Cheng Lin

(Received April 16, 2003, revised September 6, 2004)
Abstract. In this paper we define the Hardy space $H^1_{\mathcal F}(\boldsymbol{R}^n)$ associated with a family $\mathcal{F}$ of sections and a doubling measure $\mu$, where $\mathcal{F}$ is closely related to the Monge-Ampère equation. Furthermore, we show that the dual space of $H^1_{\mathcal F}(\boldsymbol{R}^n)$ is just the space $BMO_{\mathcal F}(\boldsymbol{R}^n)$, which was first defined by Caffarelli and Gutiérrez. We also prove that the Monge-Ampère singular integral operator is bounded from $H^1_{\mathcal F}(\boldsymbol{R}^n)$ to $L^1(\boldsymbol{R}^n,d \mu)$.

2000 Mathematics Subject Classification. Primary 42B30; Secondary 35B45.

Key words and phrases. $BMO$'s, Hardy spaces, Monge-Ampère equation, singular integral operators.