# Sun-Yung Alice Chang

## Princeton University

### On Moser–Trudinger–Onofri inequalty under constraints

A classical result of Aubin states that the constant in Moser–Trudinger–Onofri inequality on the 2-sphere \(S^2\) can be improved for functions with zero first order moments of the area element. In a joint work with Fengbo Hang, we generalize Aubin’s inequality to higher order moments case. These new inequalities bear similarity to a sequence of Lebedev–Milin type inequalities on the unit circle \(S^1\) coming from the work of Grenander–Szego on Toeplitz determinants (as pointed out by Widom). I will also report some joint work with Changfeng Gui still in progress, where we made attempt to formulate the analogue on \(S^2\) of the second inequality in Szego’s limit theorem.