The stability of Sharp Poincaré and Brascamp--Lieb inequalities with full monomial weights

Abstract

It was known that the Gaussian measures with monomial weights satisfy the sharp pointwise curvature-dimension condition CD(1, ∞), and consequently they allow us to derive the sharp constant $1$ for the Poincaré inequality only with partial monomial weights. In this paper, we first establish the Gaussian measures with full monomial weights satisfy the integrated curvature-dimension condition ICD(2, ∞) which is sufficient for us to prove the sharp Poincaré inequalities with optimal constant $2$ and the stability of the Brascamp--Lieb inequality for full monomial Gaussian measures. Several of these results are extended to homogeneous Gaussian measures on convex cones. Using the sharp Poincaré inequality and the hypercontractivity, we obtain an improved Beckner inequality that recover the sharp Poincaré and logarithmic Sobolev constants at the endpoints. By developing a Laguerre spectral decomposition of the associated Ornstein--Uhlenbeck type generator, we derive an exact identity for the Poincaré deficit and obtain sharp gradient stability for the Poincaré inequality with full monomial weights. We further establish L² and weighted gradient stability estimates for the Brascamp--Lieb inequality with explicit constants when all monomial exponents exceed $1$. The L² estimate extends to log-concave homogeneous weights under a subharmonicity assumption. Finally, we obtain an improved integrated curvature-dimension bound under a scale-invariant Hessian condition and identify the sharp constant for the Poincaré inequality in the radial class.

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BibTeX RIS

BibTeX

@misc{lam2026stability,
  title = {{The stability of Sharp Poincaré and Brascamp--Lieb inequalities with full monomial weights}},
  author = {Nguyen Lam and Guozhen Lu and Andrey Russanov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10337},
  eprint = {2610.10337},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10337},
}

RIS

TY  - UNPB
TI  - The stability of Sharp Poincaré and Brascamp--Lieb inequalities with full monomial weights
AU  - Nguyen Lam
AU  - Guozhen Lu
AU  - Andrey Russanov
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10337
UR  - https://arxiv.org/abs/2610.10337
AB  - It was known that the Gaussian measures with monomial weights satisfy the sharp pointwise curvature-dimension condition CD(1, ∞), and consequently they allow us to derive the sharp constant $1$ for the Poincaré inequality only with partial monomial weights. In this paper, we first establish the Gaussian measures with full monomial weights satisfy the integrated curvature-dimension condition ICD(2, ∞) which is sufficient for us to prove the sharp Poincaré inequalities with optimal constant $2$ and the stability of the Brascamp--Lieb inequality for full monomial Gaussian measures. Several of these results are extended to homogeneous Gaussian measures on convex cones. Using the sharp Poincaré inequality and the hypercontractivity, we obtain an improved Beckner inequality that recover the sharp Poincaré and logarithmic Sobolev constants at the endpoints. By developing a Laguerre spectral decomposition of the associated Ornstein--Uhlenbeck type generator, we derive an exact identity for the Poincaré deficit and obtain sharp gradient stability for the Poincaré inequality with full monomial weights. We further establish L^2 and weighted gradient stability estimates for the Brascamp--Lieb inequality with explicit constants when all monomial exponents exceed $1$. The L^2 estimate extends to log-concave homogeneous weights under a subharmonicity assumption. Finally, we obtain an improved integrated curvature-dimension bound under a scale-invariant Hessian condition and identify the sharp constant for the Poincaré inequality in the radial class.
ER  -

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