测地空间中的泛函不等式与Hamilton-Jacobi方程
Functional inequalities and Hamilton-Jacobi Equations in Geodesic Spaces
摘要：
研究了在真测地度量空间中共轭指数$p\geq 2$，$q\leq 2$的$p$-Talagrand不等式与$q$-对数Sololev不等式之间的联系.借助于一般的Hamilton-Jacobi半群，我们证明了它们是等价的，并且等价于Hamilton-Jacobi半群的超压缩性。我们的结果推广了洛特和维拉尼的结果。在Heisenberg群的亚黎曼背景下，它们可用于推导$p$ -Talagrand不等式.
英文摘要：
We study the connection between the $p$--Talagrand inequality and the $q$--logarithmic Sololev inequality for conjugate exponents $p\geq 2$, $q\leq 2$ in proper geodesic metric spaces. By means of a general Hamilton--Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton--Jacobi semigroup. Our results generalize those of Lott and Villani. They can be applied to deduce the $p$-Talagrand inequality in the sub-Riemannian setting of the Heisenberg group.
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