ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 35P15, 60J60 Wang, Feng-Yu, Department of Mathematics, Beijing Normal University, Beijing 100875, China Harnack inequality for heat semigroups and log-Sobolev inequalities A dimension free Harnack inequality is established for heat semigroups with curvature bounded below. As applications, some dimension free estimates of log-Sobolev constant are presented for compact manifolds, and furthermore, some integral conditions and differentials ones are obtained for the log-Sobolev inequality on noncompact Riemannian manifolds. Let P_{t} be a diffusion semigroup with curvature bounded below by −K, K ≥ 0. Then

[P_{t}f(x)]^{α} ≤ P_{t}f^{α}(y) exp[αKd(x, y)^{2}/2(α−1)(1 exp(2K)−1)], α > 1, f ≥ 0.

The log-Sobolev inequality holds if there exists r > 0 such that exp[(K + r)d(p, ·)^{2}] is integrable w. r. t. the invariant probability measure. Here, p is a fixed point and d is the Riemannian distance.