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 $$J_{\alpha }(u)=\frac{\alpha }{4}{\int }_{S^2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}d\omega +{\int }_{S^2}ud\omega -\ln {\int }_{S^2}{e}^{u}d\omega ,$$ 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 $\alpha \geq 1$\mathrm{.}Inthisnote,weshowthatif$J_\alpha$isrestrictedtothose$u\in H^1(S^2)$ 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., $J_{\alpha }(u)\ge 0$) whenever $\alpha \ge 2\mathrm{/}3-{ϵ}_0$ for some ${ϵ}_0>0$. The question of Chang-Yang on whether this remains true for all $\alpha \ge 1\mathrm{/}2$ remains open.