Sharp L^2 stability for anisotropic radial Caffarelli-Kohn-Nirenberg inequalities

Abstract

We establish quantitative stability for radial Caffarelli-Kohn-Nirenberg inequalities involving the anisotropic radial derivative and the Minkowski functional of a smooth, strictly convex body in ℝ^N, N≥2. Here, the convex body need not be origin-symmetric. For the L²-Caffarelli-Kohn-Nirenberg inequality, we establish the sharp stability constant |b+1−a| for a≠ b+1, and classify all equality cases in the natural weighted completion spaces, by optimizing over the angular coefficients and a common dilation parameter. We also obtain the sharp stability constants $2$ and $4$ for the scale-invariant and scale-noninvariant L²-Heisenberg inequality, respectively. For 2≤ p<∞, we also establish weighted stability of the anisotropic radial L^p-Caffarelli-Kohn-Nirenberg inequalities when a≠ b+1 and (b+1−a)(N−p(b+1))≥0, and the anisotropic radial L^p-Heisenberg inequalities. As a byproduct, we establish weighted L^p-Poincaré inequalities for Gamma measures, which serve as the main tool for our stability estimates.

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BibTeX RIS

BibTeX

@misc{wei2026sharp,
  title = {{Sharp $L^2$ stability for anisotropic radial Caffarelli-Kohn-Nirenberg inequalities}},
  author = {Zhenzhen Wei and Denghui Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40360},
  eprint = {2610.40360},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40360},
}

RIS

TY  - UNPB
TI  - Sharp L^2 stability for anisotropic radial Caffarelli-Kohn-Nirenberg inequalities
AU  - Zhenzhen Wei
AU  - Denghui Wu
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40360
UR  - https://arxiv.org/abs/2610.40360
AB  - We establish quantitative stability for radial Caffarelli-Kohn-Nirenberg inequalities involving the anisotropic radial derivative and the Minkowski functional of a smooth, strictly convex body in ℝ^N, N≥2. Here, the convex body need not be origin-symmetric. For the L^2-Caffarelli-Kohn-Nirenberg inequality, we establish the sharp stability constant |b+1−a| for a≠ b+1, and classify all equality cases in the natural weighted completion spaces, by optimizing over the angular coefficients and a common dilation parameter. We also obtain the sharp stability constants $2$ and $4$ for the scale-invariant and scale-noninvariant L^2-Heisenberg inequality, respectively. For 2≤ p<∞, we also establish weighted stability of the anisotropic radial L^p-Caffarelli-Kohn-Nirenberg inequalities when a≠ b+1 and (b+1−a)(N−p(b+1))≥0, and the anisotropic radial L^p-Heisenberg inequalities. As a byproduct, we establish weighted L^p-Poincaré inequalities for Gamma measures, which serve as the main tool for our stability estimates.
ER  -

Sharp L² stability for anisotropic radial Caffarelli-Kohn-Nirenberg inequalities

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