The stability of Sharp Poincar\'e 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 $CD(1, \infty)$, and consequently they allow us to derive the sharp constant $1$ for the Poincar\'e 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 $ICD(2, \infty)$ which is sufficient for us to prove the sharp Poincar\'e 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 Poincar\'e inequality and the hypercontractivity, we obtain an improved Beckner inequality that recover the sharp Poincar\'e 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 Poincar\'e deficit and obtain sharp gradient stability for the Poincar\'e 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 Poincar\'e inequality in the radial class.