Liouville theorem for bounded harmonic functions on manifolds and graphs satisfying non-negative curvature dimension condition

• DOI: 10.1007/s00526-019-1485-4
• Publication Date: 2019-02-10T06:02:31Z
• Authors: Bobo Hua
• Keywords: Mathematics - Functional Analysis; Mathematics - Metric Geometry; Probability (math.PR); FOS: Mathematics; Mathematics - Combinatorics; Metric Geometry (math.MG); Combinatorics (math.CO); Mathematics - Probability; Functional Analysis (math.FA)

9 pages, we added the proof of Brighton's Liouville theorem. Any comment is welcome<br/>Brighton [Bri13] proved the Liouville theorem for bounded harmonic functions on weighted manifolds satisfying non-negative curvature dimension condition, i.e. $CD(0,\infty).$ In this paper, we provide a new proof of this result by using the reverse Poincar�� inequality. Moreover, we adopt this approach to prove the Liouville theorem for bounded harmonic functions on graphs satisfying the $CD(0,\infty)$ condition.<br/>