Sharp Sobolev inequalities under first moment constraints

@misc{frank2026sharp,
  title = {{Sharp Sobolev inequalities under first moment constraints}},
  author = {Rupert L. Frank and Tianling Jin and Hua-Yang Wang and Jingang Xiong},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20001},
  eprint = {2610.20001},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20001},
}

TY  - UNPB
TI  - Sharp Sobolev inequalities under first moment constraints
AU  - Rupert L. Frank
AU  - Tianling Jin
AU  - Hua-Yang Wang
AU  - Jingang Xiong
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20001
UR  - https://arxiv.org/abs/2610.20001
AB  - Let n≥2 and 1<p<n. We study sharp Sobolev inequalities on the round sphere 𝕊^n for functions u∈ W^1,p(𝕊^n) satisfying ∫_𝕊^nx|u|^np/(n−p) dσ=0. We prove that |u|_L^np/(n−p)(𝕊^n)^p ≤ 2^−p/nS_n,p^p|∇ u|_L^p(𝕊^n)^p+B|u|_L^p(𝕊^n)^p holds for all such functions with some B=B(n,p)<∞ if and only if 1<p≤2 or 2<p<n<3p−2, where S_n,p is the optimal Euclidean Sobolev constant. This determines the exact range in which Aubin's inequality holds with the sharp gradient coefficient. For every 2≤ p<n, we also establish the corresponding inequality with squared norms and gradient coefficient 2^−2/nS_n,p^2. The gradient coefficient in each inequality is optimal. The proof combines stereographic averaging with quantitative stability of the Euclidean Sobolev inequality to obtain local second moment estimates that allow absorption of the localization errors.
ER  -