Hardy Spaces Associated with Monge-Ampere Equation

The main concern of this paper is to study the boundedness of singular integrals related to the Monge-Ampere equation established by Caffarelli and Gutierrez. They obtained the L2 boundedness. Since then the L p, 1 < p < 8, 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 H pF via the Littlewood-Paley theory with the Monge-Ampere measure satisfying the doubling property together with the noncollapsing condition, and show the H pF boundedness of Monge-Ampere singular integrals. The approach is based on the L2 theory and the main tool is the discrete Calderon reproducing formula associated with the doubling property only.