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 $L^1(\boldsymbol{R}^n,d\mu)$.
