发表机构
Memorial University of Newfoundland; University of Connecticut（纽芬兰纪念大学; 康涅狄格大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明完全单项式权重高斯测度满足ICD(2,∞)条件，从而得到最优常数2的尖锐Poincaré不等式及Brascamp--Lieb稳定性，并借助Laguerre谱分解导出精确亏损恒等式和梯度稳定性。
AI 中文摘要
已知具有单项式权重的高斯测度满足尖锐的点态曲率-维数条件$CD(1, \infty)$，因此它们允许我们仅针对部分单项式权重推导出Poincaré不等式的尖锐常数$1$。在本文中，我们首先建立具有完全单项式权重的高斯测度满足积分曲率-维数条件$ICD(2, \infty)$，这足以证明具有最优常数$2$的尖锐Poincaré不等式以及完全单项式高斯测度的Brascamp--Lieb不等式的稳定性。其中一些结果被推广到凸锥上的齐次高斯测度。利用尖锐Poincaré不等式和超收缩性，我们获得了一个改进的Beckner不等式，在端点处恢复了尖锐Poincaré常数和对数Sobolev常数。通过发展相关的Ornstein--Uhlenbeck型生成元的Laguerre谱分解，我们推导出Poincaré亏损的精确恒等式，并获得具有完全单项式权重的Poincaré不等式的尖锐梯度稳定性。我们进一步建立了当所有单项式指数超过$1$时，具有显式常数的Brascamp--Lieb不等式的$L^2$和加权梯度稳定性估计。$L^2$估计在次调和性假设下扩展到对数凹齐次权重。最后，我们在尺度不变的Hessian条件下获得了改进的积分曲率-维数界，并确定了径向类中Poincaré不等式的尖锐常数。
英文摘要
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 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 $ICD(2, \infty)$ 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.
