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Davies–Gaffney–Grigor’yan lemma on simplicial complexes

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Abstract

We prove Davies–Gaffney–Grigor’yan lemma for heat kernels of bounded discrete Hodge Laplacians on simplicial complexes.

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References

  1. Bauer, F., Hua, B., Keller, M.: On the $l^{p}$ spectrum of Laplacians on graphs. Adv. Math. 248, 717–735 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bauer, F., Hua, B., Yau, S.T.: Davies–Gaffney–Grigor’yan lemma on graphs. Commun. Anal. Geom. 23(5), 1031–1068 (2015)

    Article  MathSciNet  Google Scholar 

  3. Bauer, F., Keller, M., Wojciechoowski, R.K.: Cheeger inequalities for unbounded graph Laplacians. J. Eur. Math. Soc. 17(2), 259–271 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bauer, F., Hua, B., Yau, S.T.: Sharp Davies–Gaffney–Grigor’yan lemma on graphs. Math. Ann. 368(3–4), 1429–1437 (2017)

    Article  MathSciNet  Google Scholar 

  5. Chung, F., Grigor’yan, A., Yau, S.T.: Upper bounds for eigenvalues of the discrete and continuous Laplace operators. Adv. Math. 117(2), 165–178 (1996)

    Article  MathSciNet  Google Scholar 

  6. Coulhon, T., Sikora, A.: Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem. Proc. Lond. Math. Soc. 96(3), 507–544 (2008)

    Article  MathSciNet  Google Scholar 

  7. Davies, E.B.: Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math. 58, 99–119 (1992)

    Article  MathSciNet  Google Scholar 

  8. Davies, E.B.: Large deviations for heat kernel on graphs. J. Lond. Math. Soc. 47(2), 65–72 (1993)

    Article  MathSciNet  Google Scholar 

  9. Delmotte, T.: Parabolic Harnack inequalities and estimates of Markov chains on graphs. Rev. Math. Iberoam. 15(1), 181–232 (1999)

    Article  Google Scholar 

  10. Eckmann, B.: Harmonische Funktionen und Randwertaufgaben in einem Komplex. Comment. Math. Helv. 17, 240–255 (1945)

    Article  MathSciNet  Google Scholar 

  11. Folz, M.: Gaussian upper bounds for heat kernels of continuous time simple random walks. Electron. J. Probab. 16(62), 1693–1722 (2011)

    Article  MathSciNet  Google Scholar 

  12. Folz, M.: Volume growth and stochastic completeness of graphs. Trans. Am. Math. Soc. 366(4), 2089–211 (2014)

    Article  MathSciNet  Google Scholar 

  13. Frank, R.L., Lenz, D., Wingert, D.: Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. 266(8), 4765–4808 (2014)

    Article  MathSciNet  Google Scholar 

  14. Gaffney, M.P.: The conservation property of the heat equation on Riemannian manifolds. Commun. Pure Appl. Math. 12, 1–11 (1959)

    Article  MathSciNet  Google Scholar 

  15. Grigor’yan, A.: Integral maximum principle and its applications. Proc. R. Soc. A 124(2), 353–362 (1994)

    MathSciNet  MATH  Google Scholar 

  16. Grigor’yan, A., Huang, X., Masamune, J.: On stochastic completeness of jump processes. Math. Z. 271(3–4), 1211–1239 (2012)

    Article  MathSciNet  Google Scholar 

  17. Huang, X.: On stochastic completeness of weighted graphs. Ph.D. thesis, Bielefeld University (2011)

  18. Horak, D., Jost, J.: Spectra of combinatorial Laplace operators on simplicial complexes. Adv. Math. 244, 303–336 (2013)

    Article  MathSciNet  Google Scholar 

  19. Horak, D., Jost, J.: Interlacing inequalities for eigenvalues of discrete Laplace operators. Ann. Glob. Anal. Geom. 43(2), 177–207 (2013)

    Article  MathSciNet  Google Scholar 

  20. Hua, B., Keller, M.: Harmonic functions of general graph Laplacians. Calc. Var. Partial Differ. Equ. 51(1–2), 343–362 (2014)

    Article  MathSciNet  Google Scholar 

  21. Huang, X., Keller, M., Masamune, J., Wojciechowski, R.K.: A note on self-adjoint extensions of the Laplacian on weighted graphs. J. Funct. Anal. 265(8), 1556–1578 (2013)

    Article  MathSciNet  Google Scholar 

  22. Li, P.: Geometric Analysis, Cambridge Studies in Advanced Mathematics, vol. 134. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  23. Li, P., Yau, S.T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156(3–4), 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  24. Pang, M.M.H.: Heat kernels of graphs. J. Lond. Math. Soc. 47(2), 50–64 (1993)

    Article  MathSciNet  Google Scholar 

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Correspondence to Xin Luo.

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Supported by NSFC, Grant no. 11401106.

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Hua, B., Luo, X. Davies–Gaffney–Grigor’yan lemma on simplicial complexes. Math. Z. 290, 1041–1053 (2018). https://doi.org/10.1007/s00209-018-2051-z

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  • DOI: https://doi.org/10.1007/s00209-018-2051-z

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