Sharp improved Sobolev inequalities on ${\mathbb{S}}^n$

We prove sharp improved Sobolev inequalities on ${\mathbb{S}}^n$, $n\ge 4$, for functions whose critical density $\mathrm{\mid }u{\mathrm{\mid }}^{2n\mathrm{/}(n-2)}$ has zero barycenter, without any symmetry assumption. The Dirichlet coefficient improves by the optimal factor $2^{-2\mathrm{/}n}$, while the coefficient of the $L^2$ term remains one. Equality holds precisely for constant functions. This establishes the sharp form of Aubin’s improved Sobolev inequality in every dimension $n\ge 4$.