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

Let $S^2$ be the 2-dimensional unit sphere and let $J_α$ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $J_α(u) = \fracα{4}\int_{S^2}|\nabla u|^2 dω+ \int_{S^2} u dω-\ln \int_{S^2} e^{u} dω,$ where $dω$ denotes Lebesgue measure on $S^2$, normalized so that $\int_{S^2} dω= 1$. Onofri had established that $J_α$ is non-negative on $H^1(S^2)$ provided $α\geq 1$. In this note, we show that if $J_α$ 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., $J_α(u)\geq0$) whenever $α\geq {2/3}-ε_0$ for some $ε_0>0$. The question of Chang-Yang on whether this remains true for all $α\geq {1/2}$ remains open.

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