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On $L^\infty$ estimates for Monge–Ampère and Hessian equations on nef classes

Bin Guo, Duong H. Phong, Freid Tong and Chuwen Wang

Vol. 17 (2024), No. 2, 749–756

The PDE approach developed earlier by the first three authors for L estimates for fully nonlinear equations on Kähler manifolds is shown to apply as well to Monge–Ampère and Hessian equations on nef classes. In particular, one obtains a new proof of the estimates of Boucksom, Eyssidieux, Guedj and Zeriahi (2010) and Fu, Guo and Song (2020) for the Monge–Ampère equation, together with their generalization to Hessian equations.

Monge–Ampere equations, Hessian equations
Mathematical Subject Classification
Primary: 53C56
Secondary: 34G20
Received: 3 December 2021
Revised: 27 June 2022
Accepted: 26 July 2022
Published: 6 March 2024
Bin Guo
Department of Mathematics & Computer Science
Rutgers University
Newark, NJ
United States
Duong H. Phong
Department of Mathematics
Columbia University
New York, NY
United States
Freid Tong
Center for Mathematical Sciences and Applications
Harvard University
Cambridge, MA
United States
Chuwen Wang
Department of Mathematics
Columbia University
New York, NY
United States

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