De Ponti Nicolò, Mondino Andrea
Dipartimento di Matematica "Casorati", Università degli Studi di Pavia, Pavia, Italy.
Mathematical Institute, University of Oxford, Oxford, UK.
J Geom Anal. 2021;31(3):2416-2438. doi: 10.1007/s12220-020-00358-6. Epub 2020 Feb 14.
The goal of the paper is to sharpen and generalise bounds involving Cheeger's isoperimetric constant and the first eigenvalue of the Laplacian. A celebrated lower bound of in terms of , , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on in terms of was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is twofold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-Émery weighted) Ricci curvature bounded below by (the inequality is sharp for as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called spaces.
本文的目标是强化并推广涉及切赫等周常数和拉普拉斯算子第一特征值的界。1970年，切赫证明了关于光滑黎曼流形的一个著名的、用(\lambda_1)、(h)表示的(\lambda_1)的下界。1982年，布泽尔建立了用(h)表示的(\lambda_1)的上界（带有维数常数），2004年，勒杜将其改进（为无维数估计），适用于里奇曲率有下界的光滑黎曼流形。本文有两个目标。第一：我们强化布泽尔和勒杜得到的不等式，得到一个对于里奇曲率（在巴克利 - 埃默里加权意义下）有下界(\kappa)的空间的无维数精确布泽尔不等式（对于(\kappa = 0)，该不等式是精确的，因为在高斯空间上能取到等号）。第二：我们所有的结果在更一般的（可能非光滑的）度量测度空间中成立，这些空间的里奇曲率在合成意义下有下界，即所谓的(RCD^*(K,N))空间。
