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Equivalent properties for CD inequalities on graphs with unbounded Laplacians. (English) Zbl 1432.58020

Summary: The CD inequalities are introduced to imply the gradient estimate of Laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the \(\mathrm{CD}(K,\infty)\) and \(\mathrm{CD}(K,n)\).

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R01 PDEs on manifolds
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B50 Modular Lie (super)algebras
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References:

[1] Chung, F., Spectral Graph Theory, CBMS Regional Congerence Series in Mahtematics, 92, American Mathemeatical Society, Providence, RI,1997. · Zbl 0867.05046
[2] Hua, B. B. and Lin, Y., Stochastic completeness for graphs with curvature dimension conditions, Adv. Math., 306, 2017, 279-302. · Zbl 1364.35402 · doi:10.1016/j.aim.2016.10.022
[3] Bakery, D., Gentil, I. and Ledoux, M., Analysis and geometry of Markov diffusion operators, Number 348 in Grundlehren der Mathematischen Wissenschaften. Springer, Cham, 2014. · Zbl 1376.60002 · doi:10.1007/978-3-319-00227-9
[4] Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D. and Yau, S.-T., Li-Yau inequality on graphs, Journal of Differential Geometry, 99, 2015, 359-405. · Zbl 1323.35189 · doi:10.4310/jdg/1424880980
[5] Keller, M. and Lenz, D., Unbounded Laplacians on graphs:basic spectral properties and the heat equation, Mathematical Modelling of Natural Phenomena, 5(4), 2011, 198-224. · Zbl 1207.47032 · doi:10.1051/mmnp/20105409
[6] Keller, M. and Lenz, D., Dirichlet forms and stochastic completeness of graphs and subgraphs, Journal Fur Die Reine Und Angewandte Mathematik, 2012(666), 2012, 1074-1076. · Zbl 1252.47090 · doi:10.1515/CRELLE.2011.122
[7] Horn, P., Lin, Y., Liu, S. and Yau, S.-T., Volume doubling, Poincaré inequality and Guassian heat kernel estimate for nonnegative curvature graphs, preprint. · Zbl 1364.35402
[8] Lin, Y. and Yau, S.-T., Ricci curvature and eigenvalue estimate on locally finite graphs, Math. Res. Lett., 17(2), 2010, 343-356. · Zbl 1232.31003 · doi:10.4310/MRL.2010.v17.n2.a13
[9] Lin, Y. and Liu, S., Equivalent properties of CD inequality on graphs, preprint. · Zbl 1424.58014
[10] Yau, S.-T. and Schoen, R., Lectures on Differential Geometry, Higher Education Press, Beijing, 2014. · Zbl 0830.53001
[11] Gudmundsson, S., An Introduction to Riemann Geometry, Beijing University, Beijing, 2004.
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