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Interior \(W^{2,p}\) estimate for small perturbations to the complex Monge–AmpÈre equation

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Abstract

Let \(w_0\) be a bounded, \(C^3\), strictly plurisubharmonic function defined on \(B_1\subset {\mathbb {C}}^n\). Then \(w_0\) has a neighborhood in \(L^{\infty }(B_1)\) with the following property: for any continuous, plurisubharmonic function u in this neighborhood solving \(1-\varepsilon \le MA(u)\le 1+\varepsilon \), one has \(u\in W^{2,p}(B_{\frac{1}{2}})\), as long as \(\varepsilon >0\) is small enough depending only on n and p. This partially generalizes Caffarelli’s interior \(W^{2,p}\) estimates for real Monge–Ampère to the complex version.

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  1. Stony Brook University, Stony Brook, USA

    Jingrui Cheng & Yulun Xu

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  1. Jingrui Cheng
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  2. Yulun Xu
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Correspondence to Jingrui Cheng.

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Communicated by Laszlo Szekelyhidi.

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Cheng, J., Xu, Y. Interior \(W^{2,p}\) estimate for small perturbations to the complex Monge–AmpÈre equation. Calc. Var. 62, 231 (2023). https://doi.org/10.1007/s00526-023-02571-x

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