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Some interior regularity estimates for solutions of complex Monge–Ampère equations on a ball

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Abstract

In this paper, we consider the Dirichlet problem of a complex Monge–Ampère equation on a ball in \({\mathbb {C}}^n\). With \({\mathcal {C}}^{1,\alpha }\) (resp. \({\mathcal {C}}^{0,\alpha }\)) data, we prove an interior \({\mathcal {C}}^{1,\alpha }\) (resp. \({\mathcal {C}}^{0,\alpha }\)) estimate for the solution. These estimates are generalized versions of the Bedford–Taylor interior \({\mathcal {C}}^{1,1}\) estimate.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Chao Li, Jiayu Li & Xi Zhang

  2. AMSS, CAS, Beijing, 100080, People’s Republic of China

    Jiayu Li

Authors
  1. Chao Li
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  2. Jiayu Li
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  3. Xi Zhang
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Correspondence to Xi Zhang.

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Communicated by A. Malchiodi.

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The authors are partially supported by NSF in China No. 11625106 and 11721101. The research was partially supported by the project “Analysis and Geometry on Bundle” of Ministry of Science and Technology of the People’s Republic of China, No. SQ2020YFA070080.

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Li, C., Li, J. & Zhang, X. Some interior regularity estimates for solutions of complex Monge–Ampère equations on a ball. Calc. Var. 60, 34 (2021). https://doi.org/10.1007/s00526-020-01911-5

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