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
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, MY. & Lin, CC. 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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DOI: https://doi.org/10.1007/s12220-017-9961-6