Sharp Cheeger-Buser Type Inequalities in RCD ( K , ∞ ) Spaces.

Abstract

The goal of the paper is to sharpen and generalise bounds involving Cheeger's isoperimetric constant h and the first eigenvalue ${\lambda }_1$ of the Laplacian. A celebrated lower bound of ${\lambda }_1$ in terms of h, ${\lambda }_1\ge h^2/4$ , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on ${\lambda }_1$ in terms of h 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 $K\in R$ (the inequality is sharp for $K>0$ 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 $\mathrm{RCD}(K,\infty )$ spaces.