Hardy Spaces Associated with Monge–Ampère Equation

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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 L2 boundedness. Since then the Lp, 1 < p< ∞, 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 HFp via the Littlewood–Paley theory with the Monge–Ampère measure satisfying the doubling property together with the noncollapsing condition, and show the HFp boundedness of Monge–Ampère singular integrals. The approach is based on the L2 theory and the main tool is the discrete Calderón reproducing formula associated with the doubling property only.

Original languageEnglish
Pages (from-to)3312-3347
Number of pages36
JournalJournal of Geometric Analysis
Issue number4
StatePublished - 15 Dec 2018


  • Doubling property
  • Hardy spaces
  • Monge–Ampère equation
  • Singular integral operators


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