Hardy Spaces Associated with Monge–Ampère Equation

The main concern of this paper is to study the boundedness of singular integrals related to the Monge–Ampère equation established by Caffarelli and Gutiérrez. They obtained the L² boundedness. Since then the L^p, 1 < p< ∞, weak (1,1) and the boundedness for these operators on atomic Hardy space were obtained by several authors. It was well known that the geometric conditions on measures play a crucial role in the theory of the Hardy space. In this paper, we establish the Hardy space HFp via the Littlewood–Paley theory with the Monge–Ampère measure satisfying the doubling property together with the noncollapsing condition, and show the HFp boundedness of Monge–Ampère singular integrals. The approach is based on the L² theory and the main tool is the discrete Calderón reproducing formula associated with the doubling property only.