Hardy Spaces Associated with Monge–Ampère Equation

Abstract

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 $$L^2$$ boundedness. Since then the $$L^p, 1<p<\infty $$ , 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 $$H^p_{\mathcal F}$$ via the Littlewood–Paley theory with the Monge–Ampere measure satisfying the doubling property together with the noncollapsing condition, and show the $$H^p_{\mathcal F}$$ boundedness of Monge–Ampere singular integrals. The approach is based on the $$L^2$$ theory and the main tool is the discrete Calderon reproducing formula associated with the doubling property only.