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–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.

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Notes

  1. 1.

    The Assumption (1.2) was not explicitly stated in [18], but it follows from [18, (2.4)] and [18, p. 3094, line 6] that

    $$\begin{aligned} \mu (S(x,r))\approx \mu (B_d(x,r))\approx \mu (B_\rho (x, r))\approx r. \end{aligned}$$

    At the end of the proof of [18, Theorem 1.6], it requires \(\mu (S)\approx 2^{-k}\), where \(S=S(x, C2^{-k})\). Moreover, Theorems 5.1, 5.2, 6.2, 6.4, and 8.1 in [18] hold under the Assumption (1.2).

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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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Han, Y., Lee, M. & Lin, C. Hardy Spaces Associated with Monge–Ampère Equation. J Geom Anal 28, 3312–3347 (2018). https://doi.org/10.1007/s12220-017-9961-6

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Keywords

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

Mathematics Subject Classification

  • 42B20
  • 42B35