Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 6, pp 2207–2225 | Cite as

Gradient estimates on connected graphs with the \(CD\psi (m,K)\) condition

  • Ying Lv
  • LinFeng WangEmail author


Let G(VE) be a finite or infinite (locally finite) graph with the \(CD\psi (m,-K)\) condition for some constants \(m>0,K\ge 0\), and some \(C^1\), concave function \(\psi : (0,+\infty )\rightarrow {\mathbf {R}}\). In this paper, we firstly prove that if \(u:V\rightarrow (0,+\infty )\) is a solution to the equation \(\triangle _{\mu } u=-\lambda u\), then \(\lambda \le \frac{mK}{4\psi '(1)}\), and u satisfies an elliptic gradient estimate. We can recover a result derived in Wang and Zhou (Archiv der Mathematik 109:383–391, 2017) if we choose \(\psi =\sqrt{\cdot }\). Secondly, we establish a general gradient estimate for the linear heat equation \((\triangle _{\mu }-\partial _t) u=0\); this estimate includes the Davies, Hamilton, Bakry–Qian and Li–Xu’s estimates; these four type estimates had been established on the manifold. We can derive gradient estimates on a finite graph with the same types as in the manifold case if we choose \(\psi =\ln {(\cdot )}\). As by-products, we get corresponding Harnack inequalities.


Graph \(CD\psi (m{, }K)\) condition Gradient estimate Harnack inequality 

Mathematics Subject Classification

53C21 05C99 



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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of ScienceNantong UniversityNantongChina

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