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New Sharp Inequalities in Analysis and Geometry

2021年6月12日9:30-10:30  闵行数学楼102报告厅

*主持人：叶东 教授

*讲座内容简介：

The classical Moser-Trudinger inequality is a borderline case of Soblolev inequalities and plays an important role in geometric analysis. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space $H^1$, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without constraints. The main ingredient leading to the above inequalities is a novel geometric inequality: Sphere Covering Inequality, discovered jointly with Amir Moradifam from UC Riverside.

One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Aubin on the sphere which is for functions with mass centeredat the origin. In another view point, this inequality also generalizes to the sphere the Lebedev-Milin inequality and the second inequality in the Szeg\"o limit theorem on the Toeplitz determinants on the circle, which is useful in the study of isospectral compactness for metrics defined on compact surfaces, among other applications.

Efforts have also been made to show similar inequalities in higher dimensions. Among the preliminary results, we have improved Beckner's inequality for axially symmetric functions when the dimension $n=4, 6, 8$. Many questions remain open.

*主讲人简介：