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We prove Aubin's "Hypothese fondamentale" concerning the existence ofMoser-Trudinger type inequalities on any integral compact K\"ahler manifold X.In the case of the anti-canonical class on a Fano manifold the constants in theinequalities are shown to only depend on the dimension of X (but there arecounterexamples to the precise value proposed by Aubin). In the differentsetting of pseudoconvex domains in complex space we also obtain a quasi-sharpversion of the inequalities and relate it to Brezis-Merle type inequalities.The inequalities are shown to be sharp for S^{1}-invariant functions on theunit-ball. We give applications to existence and blow-up of solutions tocomplex Monge-Amp\`ere equations of mean field (Liouville) type.

Moser-Trudinger type inequalities for complex Monge-Amp\`ere operators and Aubin's "hypoth\`ese fondamentale"