University of Texas at San Antonio |

**Title:** New Sharp Inequalities in Analysis and Geometry**Abstract:** 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 centered at the origin. In

another view point, this inequality also generalizes to the sphere the

Lebedev-Milin inequality and the second inequality in the Szegö 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.

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