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

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.