Abstract
We prove a local regularity (and a corresponding a priori estimate) for plurisubharmonic solutions of the nondegenerate complex Monge–Ampère equation assuming that their W 2, p-norm is under control for some p > n(n − 1). This condition is optimal. We use in particular some methods developed by Trudinger and an estimate for the complex Monge–Ampère equation due to Kołodziej.
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Acknowledgments
Part of the research was done while the second named author was visiting the Princeton University. He would like to thank this institution for the perfect working conditions and hospitality and especially professor Gang Tian for his encouragement and help.
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Partially supported by the projects N N201 2683 35 and 189/6 PR EU/2007/7 of the Polish Ministry of Science and Higher Education.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Błocki, Z., Dinew, S. A local regularity of the complex Monge–Ampère equation. Math. Ann. 351, 411–416 (2011). https://doi.org/10.1007/s00208-010-0609-0
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DOI: https://doi.org/10.1007/s00208-010-0609-0