Perpetual cutoff method and CD{E}^{\prime }(K,N) condition on graphs

Abstract

By using the perpetual cutoff method, we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CD{E}^{\prime }(K,N). This generalizes a main result of F. Münch who considers the case of CD(K, \infty) curvature. Hence, we answer a question raised by Münch. For that purpose, we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded Laplacian \text{Δ} and perpetual cutoff semigroup P_t^W in our setting.

Key words： Locally finite graphs; perpetual cutoff method; gradient estimates; CD{E}^{\prime }(K,N)

References

1 Bauer F, Horn P, Lin Y, Lippner G, Mangoubi D, Yau S T. Li-Yau inequality on graphs. J Differential Geom, 2015, 99(3): 359–405

2 Erbar M, Fathi M. Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature. J Funct Anal, 2018, 274(11): 3056–3089

3 Gong C, Lin Y. Equivalent properties for CD inequalities on graphs with unbounded Laplacians. Chin Ann Math Ser B, 2017, 38(5): 1059–1070

4 Gong C, Lin Y, Liu S, Yau S T. Li-Yau inequality for unbounded Laplacian on graphs. Adv Math, 2019, 357: 1–23

5 Hagood J W, Thomson B S. Recovering a function from a Dini derivative. Amer Math Monthly, 2006, 113(1): 34–46

6 Horn P. A spacial gradient estimate for solutions to the heat equation on graphs. SIAM J Discrete Math, 2019, 33(2): 958–975

7 Horn P, Lin Y, Liu S, Yau S T. Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs. J Reine Angew Math, 2019, 757: 89–130

8 Hua B B, Lin Y. Stochastic completeness for graphs with curvature dimension conditions. Adv Math, 2017, 306: 279–302

9 Keller M, Lenz D. Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math Model Nat Phenom, 2010, 5(4): 198–224

10 Keller M, Münch F. Gradient estimates, Bakry-Emery Ricci curvature and ellipticity for unbounded graph Laplacians. arXiv: 1807.10181v1

11 Kempton K, Lippner G, Münch F. Large scale Ricci curvature on graphs. Calc Var Partial Differential Equations, 2020, 59(5): 166–17 pp)

12 Lin Y, Liu S. Equivalent properties of CD inequality on graph. arXiv: 1512.02677v1

13 Lin Y, Yau S T. Ricci curvature and eigenvalue estimate on locally finite graphs. Math Res Lett, 2010, 17(2): 343–356

14 Münch F. Remarks on curvature dimension conditions on graphs. Calc Var Partial Differential Equations, 2017, 56(1): 11–8 pp)

15 Münch F. Li-Yau inequality on finite graphs via non-linear curvature dimension conditions. J Math Pures Appl, 2018, 120: 130–164

16 Münch F. Perpetual cutoff method and discrete Ricci curvature bounds with exceptions. arXiv: 1812.02593v1

17 Münch F. Li-Yau inequality under CD(0,N) on graphs. arXiv: 1909.10242v1

18 Münch F, Wojciechowski R K. Ollivier Ricci curvature for general graph Laplacian: heat equation, Laplacian comparison, non-explosion and diameter bounds. Adv Math, 2019, 356: 106759 (45 pp)