Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter
Let \(\rho_0\) be the logarithmic Sobolev constant of the Laplacian on a Riemannian manifold \(M\) with finite (normalized) Riemannian measure \(\mu\). Using a simple differential inequality due to I. Herbst, E. B. Davies and B. Simon, the author presents short and simple proofs of exponential integrability of Lipschitz functions and bounds on the diameter \(D\) of \(M\) provided \(\rho_0 > 0\). It is shown in particular that if the Ricci curvature of \(M\) is bounded below, then \(D \leq c/ \min (1, \rho_0)\), where \(C\) only depends on the dimension of \(M\) and the bound on the Ricci curvature. The bound is sharp and improves upon a recent result of \textit{L. Saloff-Coste} [Colloq. Math. 67, No. 1, 109-121 (1994; Zbl 0816.53027)].
scientific article; zbMATH DE number 836411