We prove sharp improved Sobolev inequalities on $\mathbb{S}^n$, $n\ge 4$, for functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter, without any symmetry assumption. The Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ term remains one. Equality holds p...
We prove sharp improved Sobolev inequalities on $\mathbb S^n$, $n\ge3$, for antipodally even functions and for axially symmetric functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter. In both cases, the Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ t...
We prove sharp residual stability estimates for the critical Poincar\'e-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, whe...
