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Attainment of the sharp stability constant of Beckner conformal log-Sobolev inequality

  • Lu Chen
  • Bohan Wang
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Abstract

The existence of minimizer for stability of the fractional Sobolev inequality and the stability of Beckner conformal Log-Sobolev inequality (the endpoint case of fractional Sobolev inequality) have been established by König and Chen-Lu-Tang. The existence of minimizer for stability of the Log-Sobolev inequality is unsolved. In this paper, we solve this problem. For every n≥2, we prove that the sharp L2-stability constant of Beckner conformal log-Sobolev inequality on 𝕊n is attained and is strictly less than 2log2/n. Moreover, the normalized stability constant of the fractional Sobolev inequality converges to the stability constant of the log-Sobolev inequality as s→0+. After Hersch normalization, every sequence of Lpsk-normalized minimizers for stability of the fractional Sobolev inequality, with sk→0+, has a subsequence converging strongly in L2(𝕊n) to an L2-normalized minimizer of stability of the log-Sobolev inequality.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026attainment,
  title = {{Attainment of the sharp stability constant of Beckner conformal log-Sobolev inequality}},
  author = {Lu Chen and Bohan Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18327},
  eprint = {2610.18327},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18327},
}
RIS
TY  - UNPB
TI  - Attainment of the sharp stability constant of Beckner conformal log-Sobolev inequality
AU  - Lu Chen
AU  - Bohan Wang
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18327
UR  - https://arxiv.org/abs/2610.18327
AB  - The existence of minimizer for stability of the fractional Sobolev inequality and the stability of Beckner conformal Log-Sobolev inequality (the endpoint case of fractional Sobolev inequality) have been established by König and Chen-Lu-Tang. The existence of minimizer for stability of the Log-Sobolev inequality is unsolved. In this paper, we solve this problem. For every n≥2, we prove that the sharp L^2-stability constant of Beckner conformal log-Sobolev inequality on 𝕊^n is attained and is strictly less than 2log2/n. Moreover, the normalized stability constant of the fractional Sobolev inequality converges to the stability constant of the log-Sobolev inequality as s→0^+. After Hersch normalization, every sequence of L^p_s_k-normalized minimizers for stability of the fractional Sobolev inequality, with s_k→0^+, has a subsequence converging strongly in L^2(𝕊^n) to an L^2-normalized minimizer of stability of the log-Sobolev inequality.
ER  -