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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3312–3347 | Cite as

Hardy Spaces Associated with Monge–Ampère Equation

  • Yongshen Han
  • Ming-Yi LeeEmail author
  • Chin-Cheng Lin
Article
  • 143 Downloads

Abstract

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 \(L^2\) 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–Ampère measure satisfying the doubling property together with the noncollapsing condition, and show the \(H^p_{\mathcal F}\) 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.

Keywords

Doubling property Hardy spaces Monge–Ampère equation Singular integral operators 

Mathematics Subject Classification

42B20 42B35 

Notes

Acknowledgements

Ming-Yi Lee and Chin-Cheng Lin are supported by the Ministry of Science and Technology, R.O.C. under Grant Nos. #MOST 106-2115-M-008-003-MY2 and #MOST 106-2115-M-008-004-MY3, respectively, as well as supported by the National Center for Theoretical Sciences of Taiwan.

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsAuburn UniversityAuburnUSA
  2. 2.Department of MathematicsNational Central University320Taiwan, ROC

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