On the Best Constant in the Moser-Onofri-Aubin Inequality

Let $S^2$ be the 2-dimensional unit sphere and let $J_\alpha$ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$J_\alpha(u) = \frac{\alpha}{4}\int_{S^2}|\nabla u|^2 d\omega + \int_{S^2} u d\omega -\ln \int_{S^2} e^{u} d\omega,$$ where $d\omega$ denotes Lebesgue measure on $S^2$, normalized so that $\int_{S^2} d\omega = 1$. Onofri had established that $J_\alpha$ is non-negative on $H^1(S^2)$ provided $\alpha \geq 1$. In this note, we show that if $J_\alpha$ is restricted to those $u\in H^1(S^2)$ that satisfy the Aubin condition: \int_{S^2}e^u x_j dw=0\quad\text{for all}1\leq j\leq 3, then the same inequality continues to hold (i.e., $J_\alpha (u)\geq0$) whenever $\alpha \geq {2/3}-\epsilon_0$ for some $\epsilon_0>0$. The question of Chang-Yang on whether this remains true for all $\alpha \geq {1/2}$ remains open.