A nonlinear inequality of Moser-Trudinger type
Let \((M,\omega)\) be a compact \(n\)--dimensional Kähler--Einstein manifold with positive scalar curvature such that \(\text{Ric}(\omega)=\omega\). Put \(V:=\int_M\omega^n\), \(J_\omega(\varphi):=\frac 1V ds \int_0^1\int_M \varphi(\omega^n-(\omega+s\partial\overline\partial\varphi)^n)\). Let \(\Lambda_1\) denote the space of all eigenfunctions of \(\omega\) with eigenvalue \(1\). The main result of the paper is the following theorem. For any almost plurisubharmonic function \(\varphi\) on \(M\) such that \(\int_M\varphi\psi\omega^n=0\), \(\psi\in\Lambda_1\), we have \(\frac 1V\int_Me^{-\varphi}\omega^n\leq C\exp(J_\omega(\varphi)- \int_M\varphi\omega^n-J_\omega^\delta(\varphi))\), where \(\delta=\delta(n)>0\), \(C=C(n,\lambda_2(\omega)-1)\), and \(\lambda_2(\omega)\) denotes the least eigenvalue of \(\omega\) which is bigger than \(1\). A weaker version of the above inequality (under some additional conditions on \(\varphi\)) was proved by the first author in [Invent. Math. 130, No. 1, 1-37 (1997; Zbl 0892.53027)]. In the case where \(M=S^2\) is the two dimensional sphere the result generalizes the so called Moser-Trudinger-Onofri inequality proved by \textit{T. Aubin} [J. Funct. Anal. 57, 143-153 (1984; Zbl 0538.53063)].
scientific article; zbMATH DE number 1484723