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.
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