Moser–Trudinger type inequalities for complex Monge–Ampère operators and Aubin’s “hypothèse fondamentale”

Robert J. Berman; Bo Berndtsson

doi:10.5802/afst.1704

Robert J. Berman ¹; Bo Berndtsson ¹

¹ Mathematical Sciences - Chalmers University of Technology and University of Gothenburg - SE-412 96 Gothenburg, Sweden

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 3, pp. 595-645.

Abstract

We prove a version of Aubin’s “Hypothèse fondamentale” concerning the existence of Moser–Trudinger type inequalities on any integral compact Kähler manifold $X$ . In the case of the anti-canonical class on a Fano manifold the constants in the inequalities are shown to only depend on the dimension of $X$ (but there are counterexamples to the precise value proposed by Aubin). In the different setting of pseudoconvex domains in complex space we also obtain a quasi-sharp version of the inequalities and relate it to Brezis–Merle type inequalities for the complex Monge–Ampère operator, recently considered by Demailly and Åhag–Cegrell–Kołodziej–Phạm–Zeriahi. The inequalities are shown to be sharp for $S^{1}$ -invariant functions on the unit ball.

Published online: 2022-07-01

DOI: 10.5802/afst.1704

Author's affiliations:

Robert J. Berman ¹; Bo Berndtsson ¹

¹ Mathematical Sciences - Chalmers University of Technology and University of Gothenburg - SE-412 96 Gothenburg, Sweden

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Robert J. Berman and Bo Berndtsson},
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Robert J. Berman; Bo Berndtsson. Moser–Trudinger type inequalities for complex Monge–Ampère operators and Aubin’s “hypothèse fondamentale”. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 3, pp. 595-645. doi : 10.5802/afst.1704. https://afst.centre-mersenne.org/articles/10.5802/afst.1704/

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