Work

Curvature and other local inequalities in Markov semigroups

Year: 2026

Type: article

Abstract: We investigate the relationship between curvature bounds and local functional inequalities. In particular, we show that local inequalities derived through the maximum principle approach of Ivanisvili and Volberg (In 50 Years with Hardy Spaces (2018) 281–305, Birkhäuser/Springer) are equivalent to lower bounds on the Ricci curvature in the sense of Bakry–Emery. Moreover, we will develop this Euclidean space technique to metric measure spaces satisfying the RCD condition, providing a unified approach to functional and isoperimetric inequalities in nonsmooth spaces with a synthetic Ricci curvature bound. Finally, we are interested in commutation properties for semigroup operators on Rn in the absence of positive curvature, based on a local eigenvalue criteria.

Source: The Annals of Probability

DOI: 10.1214/25-aop1779

Authors: Devraj Duggal, Andreas Malliaris, James Melbourne, Cyril Roberto

Institutions: University of Minnesota, Université de Toulouse, Institut de Mathématiques de Toulouse, Mathematics Research Center, Université Paris Nanterre

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Agence Nationale de la Recherche

Awards: G4656594441, Mathematics and Interactions in Toulouse