论文标题
急剧加权的日志贝贝尔不等式：平等案例和应用的表征
Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications
论文作者
论文摘要
通过使用最佳的质量传输理论，我们为尖锐的$ l^p $ -log-sobolev不平等$（p \ geq 1）$提供了直接证明，该$涉及log-conconcave均匀重量，以开放的凸锥$ e \ subseteq \ subseteq \ subseteq \ mathbb r^n $。此证明的津贴是，它允许表征$ l^p $ -log-sobolev不平等中相等案例的极端功能。即使在未加权的设置和$ e = \ Mathbb r^n $中，平等案例的表征对于$ p \ geq n $也是新的。作为一种应用，我们为与汉密尔顿 - 雅各比方程相关的HOPF-LAX半群提供了急剧的加权超收缩率，这也表征了平等案例。
By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the $L^p$-log-Sobolev inequality. The characterization of the equality cases is new for $p\geq n$ even in the unweighted setting and $E=\mathbb R^n$. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.
