On Dolbeault's conjecture

We prove sharp improved Sobolev inequalities on ${\mathbb{S}}^n$, $n\ge 3$, for antipodally even functions and for axially symmetric functions whose critical density $\mathrm{\mid }u{\mathrm{\mid }}^{2n\mathrm{/}(n-2)}$ has zero barycenter. In both cases, 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. The even case resolves a conjecture of Dolbeault (2021).