Local Poincaré inequalities from stable curvature conditions on metric spaces (Q425162)

A classical result of \textit{P. Buser} [Ann. Sci. Éc. Norm. Supér. (4) 15, 213--230 (1982; Zbl 0501.53030)] implies that manifolds of nonnegative Ricci curvature satisfy a local \(L^1\) Poincaré inequality. The paper under review discusses generalisations of this result to metric measure spaces. First, there is the class of \(CD(0,N)\) spaces introduced by \textit{J. Lott} and \textit{C. Villani} in [Ann. Math. (2) 169, No. 3, 903--991 (2009; Zbl 1178.53038)]. These are defined by the condition that any two probability measures can be joined by a Wasserstein geodesic along which certain entropy functionals are convex, and they generalize N-dimensional Riemannian manifolds of nonnegative Ricci curvature. In the paper under review it is shown that any \(CD(0,\infty)\) space satisfies a weak local \(L^1\) Poincaré inequality and that any \(CD(0,N)\) space satisfies a uniform weak local \(L^1\) Poincaré inequality. This generalizes former results of Lott-Villani [loc. cit.] and \textit{M.-K. von Renesse} [Math. Z. 259, No. 1, 21--31 (2008; Zbl 1141.53076)]. Second, there is the class of spaces with a flow satisfying the evolution variational inequality (E.V.I.) with respect to the Rényi entropy \(\mathcal{E}_N\) or the Shannon entropy \(\mathcal{E}_\infty\). This condition implies convexity along Wasserstein geodesics of the entropy functional in the sense of Sturm, but not necessarily in the sense of Lott-Villani. In the paper under review it is shown that convexity of \(\mathcal{E}_N\) or \(\mathcal{E}_\infty\) along any Wasserstein geodesic also implies a weak local \(L^1\) Poincaré inequality. The results of the paper under review were used by \textit{R. Jiang} and \textit{P. Koskela} [Commun. Pure Appl. Math. 65, No. 8, 1145--1168 (2012; Zbl 1247.53045)] to derive isoperimetric inequalities and local Sobolev inequalities in Ahlfors-regular metric measure spaces.