Weak Poincaré Inequalities in the Absence of Spectral Gaps

Ben-Artzi, Jonathan; Einav, Amit

Abstract

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by Liggett (Ann Probab 19(3):935–959, 1991). Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup.

Citation

Ben-Artzi, J., & Einav, A. (2020). Weak Poincaré Inequalities in the Absence of Spectral Gaps. Annales Henri Poincaré, 21(2), 359–375. https://doi.org/10.1007/s00023-019-00858-4

• Journal Article Type: Article
• Acceptance Date: Oct 9, 2019
• Online Publication Date: Oct 29, 2019
• Publication Date: 2020-02
• Deposit Date: Nov 16, 2020
• Journal: Annales Henri Poincaré
• Print ISSN: 1424-0637
• Electronic ISSN: 1424-0661
• Publisher: Springer
• Volume: 21
• Issue: 2
• Pages: 359–375
• DOI: https://doi.org/10.1007/s00023-019-00858-4
• Public URL: https://durham-repository.worktribe.com/output/1257225