A PDE proof is provided for the sharp $L^\infty$ estimates for the complex Monge-Ampère equation that had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics. It extends to more general fully non-linear equations satisfying a structural condition, and it also gives estimates of Trudinger type.
Ann. of Math. (2)
198(1):
393-418
(July 2023).
DOI: 10.4007/annals.2023.198.1.4
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