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

Juncheng Wei; Zetian Yan; Zikai Ye

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arXiv · 2610.23069

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

Juncheng Wei·Zetian Yan·Zikai Ye

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Abstract

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 precisely for constant functions. This establishes the sharp form of Aubin's improved Sobolev inequality in every dimension $n\ge4$.

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Juncheng Wei, Zetian Yan, Zikai Ye. 2026-10-16. Sharp improved Sobolev inequalities on $\mathbb{S}^n$. https://arxiv.org/abs/2610.23069

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