LOG-SOBOLEV INEQUALITIES: DIFFERENT ROLES OF RIC AND HESS

• 成果类型: Article
• 署名作者: Wang, Feng-Yu
• 署名单位: Beijing Normal University; Swansea University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP444
• 发表日期: 2009
• 页码: 1587-1604
• 关键词: riemannian-manifolds; heat kernel; ultracontractivity; semigroups; PROPERTY

摘要

Let P(t) be the diffusion semigroup generated by L := Delta + del V on a complete connected Riemannian manifold with Ric >= -(sigma(2)rho(2)(0) + c) for some constants sigma, c > 0 and rho(0) the Riemannian distance to a fixed point. It is shown that Pt is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided - Hess(V) >= delta holds outside of a compact set for some constant delta > (1 +root 2)sigma root d-1. This indicates, at least in finite dimensions, that Ric and - HessV play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.