论文标题
带有应用
Sharp log-Sobolev inequalities in ${\sf CD}(0,N)$ spaces with applications
论文作者
论文摘要
给定的$ p，n> 1，$我们证明了在非策略度量度量空间上的尖锐$ l^p $ -log-sobolev不平等，满足$ {\ sf cd}（0，n）$条件，其中最佳常数涉及该空间的渐近量比。此证明基于$ {\ sf cd}（0，n）$空间，对称性和仔细的缩放参数的尖锐等级不等式。作为一个应用程序，我们为$ {\ sf cd}（0，n）$ spaces建立了HOPF-LAX SEMIGROUP的尖锐超收缩率。此结果的证据使用了Hopf-Lax半群的Hamilton-Jacobi不平等和SOBOLEV的规律性特性，事实证明这在当前非平滑和非划界空间的情况下至关重要。此外，还以$ {\ sf rcd}（0，n）$ spaces获得了尖锐的高斯型$ l^2 $ -log-sobolev不平等。我们的结果是新的，即使在Riemannian/Finsler歧管的平稳环境中也是如此。特别是，ni（J.Geom。Anal。，2004）对Riemannian歧管的著名刚性结果的扩展将是我们敏锐的log-Sobolev不平等的简单结果。
Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in ${\sf CD}(0,N)$ spaces, symmetrisation, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in ${\sf CD}(0,N)$ spaces. The proof of this result uses Hamilton-Jacobi inequality and Sobolev regularity properties of the Hopf-Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Furthermore, a sharp Gaussian-type $L^2$-log-Sobolev inequality is also obtained in ${\sf RCD}(0,N)$ spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (J. Geom. Anal., 2004) on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.
