Sharp Sobolev inequalities under first moment constraints

Let $n\ge 2$ and $1<p<n$. We study sharp Sobolev inequalities on the round sphere ${\mathbb{S}}^n$ for functions $u\in W^{1,p}({\mathbb{S}}^n)$ satisfying ${\int }_{{\mathbb{S}}^n}x\mathrm{\mid }u{\mathrm{\mid }}^{np\mathrm{/}(n-p)}d\sigma =0$. We prove that $$\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{np\mathrm{/}(n-p)}({\mathbb{S}}^n)}^p\le 2^{-p\mathrm{/}n}{S}_{n,p}^p\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p({\mathbb{S}}^n)}^p+B\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p({\mathbb{S}}^n)}^p$$ holds for all such functions with some $B=B(n,p)<\mathrm{\infty }$ if and only if $1<p\le 2$ or $2<p<n<3p-2$, where $S_{n,p}$ is the optimal Euclidean Sobolev constant. This determines the exact range in which Aubin’s inequality holds with the sharp gradient coefficient. For every $2\le p<n$, we also establish the corresponding inequality with squared norms and gradient coefficient $2^{-2\mathrm{/}n}{S}_{n,p}^2$. The gradient coefficient in each inequality is optimal. The proof combines stereographic averaging with quantitative stability of the Euclidean Sobolev inequality to obtain local second moment estimates that allow absorption of the localization errors.