Hardy spaces associated to the sections

Abstract

In this paper we define the Hardy space $H^1_{\Cal F}(\Bbb R^n)$ associated with a family $\Cal {F}$ of sections and a doubling measure $\mu$, where $\Cal {F}$ is closely related to the Monge-Ampère equation. Furthermore, we show that the dual space of $H^1_{\Cal F}(\Bbb R^n)$ is just the space $B\!M\!O_{\Cal F}(\Bbb R^n)$, which was first defined by Caffarelli and Gutiérrez. We also prove that the Monge-Ampère singular integral operator is bounded from $H^1_{\Cal F}(\Bbb R^n)$ to $L^1(\Bbb R^n,d\mu)$.