PreprintCC BY 4.0arXiv
Sharp improved Sobolev inequalities on 𝕊n
Abstract
We prove sharp improved Sobolev inequalities on 𝕊n, n≥ 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 L2 term remains one. Equality holds precisely for constant functions. This establishes the sharp form of Aubin's improved Sobolev inequality in every dimension n≥4.
Related
- On improved Hardy-type inequalities in Log-Weighted Sobolev Spaces
- Sharp Sobolev inequalities on minimal immersions via optimal transport
- Sharp logarithmic Sobolev inequalities on free groups by variational comparison
- Sharp Logarithmic Sobolev Inequalities for q-Araki-Woods Ornstein-Uhlenbeck Semigroups
- Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals
BibTeX
@misc{wei2026sharp,
title = {{Sharp improved Sobolev inequalities on $\mathbbS^n$}},
author = {Juncheng Wei and Zetian Yan and Zikai Ye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23069},
eprint = {2610.23069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23069},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp improved Sobolev inequalities on 𝕊^n
AU - Juncheng Wei
AU - Zetian Yan
AU - Zikai Ye
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23069
UR - https://arxiv.org/abs/2610.23069
AB - We prove sharp improved Sobolev inequalities on 𝕊^n, n≥ 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≥4.
ER -