Hardy spaces associated with Monge-Ampère equation

Chin-Cheng Lin

Department of Mathematics

National Central University

clin@math.ncu.edu.tw

    We study the boundedness of singular integrals related to the Monge-Ampère equation established by Caffarelli and Gutiérrez. They obtained the `L^2` boundedness. Since then the `L^p,1\lt p\lt\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 talk, we establish the Hardy space `H_\mathcal{F}^p` via the Littlewood-Paley theory with the Monge-Ampère measure satisfying the doubling property together with the noncollapsing condition, and show the `H_{\mathcal{F}}^p` boundedness of Monge-Ampère singular integrals. The approach is based on the `L^2` theory and the main tool is the discrete Calderón reproducing formula associated with the doubling property only.