A lower bound for the stability constant in the Caffarelli--Kohn--Nirenberg inequality

Abstract

We study the dependence on the dimension of stability constants for the Caffarelli--Kohn--Nirenberg inequalities, using the weighted energy distance to the manifold of optimizers. In the subcritical non-degenerate case, i.e., b_FS(a)<b<a+1 with a<0 and a<b<a+1 with a≥0, we prove β_a,b/d≤ c_d,a,b ≤ (4(1+a−b))/(d+4(1+a−b)) with a positive constant depending only on the weights, uniformly over all admissible dimensions. The upper bound shows that this dimensional order is optimal. The critical case a=b>0 exhibits a different dimensional scale, min{β_Sob,4a}/25d² ≤ c_d,a,a≤16a/d²+O_a(d⁻³), where β_Sob>0 is a universal Sobolev stability constant. In the degenerate case on the Felli--Schneider curve, for fixed a<0, we prove β_a^FS/d³ ≤c̃_{d,a,b_FS(a)} ≤32|a|/d³+O_a(d⁻⁴), where c̃ is the quartic stability constant given in [Frank-Peteranderl, CV\&PDE, 2024] and β_a^FS>0 is independent of d. The three cases thus have optimal dimensional orders d⁻¹, d⁻², and d⁻³, respectively.

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BibTeX RIS

BibTeX

@misc{li2026lower,
  title = {{A lower bound for the stability constant in the Caffarelli--Kohn--Nirenberg inequality}},
  author = {Houwang Li and Juncheng Wei},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41474},
  eprint = {2610.41474},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41474},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A lower bound for the stability constant in the Caffarelli--Kohn--Nirenberg inequality
AU  - Houwang Li
AU  - Juncheng Wei
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41474
UR  - https://arxiv.org/abs/2610.41474
AB  - We study the dependence on the dimension of stability constants for the Caffarelli--Kohn--Nirenberg inequalities, using the weighted energy distance to the manifold of optimizers. In the subcritical non-degenerate case, i.e., b_FS(a)<b<a+1 with a<0 and a<b<a+1 with a≥0, we prove β_a,b/d≤ c_d,a,b ≤ (4(1+a−b))/(d+4(1+a−b)) with a positive constant depending only on the weights, uniformly over all admissible dimensions. The upper bound shows that this dimensional order is optimal. The critical case a=b>0 exhibits a different dimensional scale, min{β_Sob,4a}/25d^2 ≤ c_d,a,a≤16a/d^2+O_a(d^−3), where β_Sob>0 is a universal Sobolev stability constant. In the degenerate case on the Felli--Schneider curve, for fixed a<0, we prove β_a^FS/d^3 ≤c̃_d,a,b_FS(a) ≤32|a|/d^3+O_a(d^−4), where c̃ is the quartic stability constant given in [Frank-Peteranderl, CV\&PDE, 2024] and β_a^FS>0 is independent of d. The three cases thus have optimal dimensional orders d^−1, d^−2, and d^−3, respectively.
ER  -

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