A nonlinear inequality of Moser-Trudinger type
type: Item
instance of: scholarly article
title: A nonlinear inequality of Moser-Trudinger type
zbMATH Open document ID: 0987.32017
author: Xiaohua Zhu
author: Gang Tian
published in: Calculus of Variations and Partial Differential Equations
publication date: 2002-06-16T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32Q20
zbMATH DE Number: 1484723
zbMATH Keywords: almost plurisubharmonic function
zbMATH Keywords: Moser-Trudinger-Onofri inequality
reviewed by: Marek Jarnicki
MaRDI profile type: Publication
full work available at URL: s005260010349
OpenAlex ID: W1981874541
DOI: 10.1007/S005260010349
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description: scientific article; zbMATH DE number 1484723