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arXiv · 2610.14359

On Dolbeault's conjecture

Juncheng Wei·Zetian Yan·Zikai Ye
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Abstract

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 $L^2$ term remains one. Equality holds precisely for constant functions. The even case resolves a conjecture of Dolbeault (2021).

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Juncheng Wei, Zetian Yan, Zikai Ye. 2026-10-09. On Dolbeault's conjecture. https://arxiv.org/abs/2610.14359

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