no. 1
Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings
[Version discrète d’une preuve de Dungey pour les estimations du gradient du noyau de la chaleur sur les revêtements]
Ishiwata, Satoshi 1
1 Institute of Mathematics University of Tsukuba 1-1-1 Tennoudai, 305-8571 Ibaraki JAPAN
Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 93-102.

We obtain another proof of a Gaussian upper estimate for a gradient of the heat kernel on cofinite covering graphs whose covering transformation group has a polynomial volume growth. It is proved by using the temporal regularity of the discrete heat kernel obtained by Blunck [2] and Christ [3] along with the arguments of Dungey [7] on covering manifolds.

DOI : 10.5802/ambp.229
Classification : 60J10, 58J35, 58J37
Mots clés : Gradient estimates, Random walks, Gaussian estimates for the heat kernel
Ishiwata, Satoshi  1

1 Institute of Mathematics University of Tsukuba 1-1-1 Tennoudai, 305-8571 Ibaraki JAPAN 
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Ishiwata, Satoshi . Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings. Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 93-102. doi : 10.5802/ambp.229. http://archive.numdam.org/articles/10.5802/ambp.229/

[1] Auscher, P.; Coulhon, T.; Duong, X. T.; Hofmann, S. Riesz transform on manifolds and heat kernel regurality, Ann. Scient. Éc. Norm. Sup., Volume 37 (2004), pp. 911-957 | Numdam | MR | Zbl

[2] Blunck, S. Perturbation of analytic operators and temporal regularity, Colloq. Math., Volume 86 (2000), pp. 189-201 | MR | Zbl

[3] Christ, M. Temporal regularity for random walk on discrete nilpotent groups, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl., Volume Special Issue (1995), pp. 141-151 | MR | Zbl

[4] Coulhon, T.; Duong, X. T. Riesz transforms for 1≤p≤2, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 1151-1169 | DOI | MR | Zbl

[5] Davies, E. B. Non-gaussian aspects of heat kernel behaviour, J. London Math. Soc., Volume 55 (1997), pp. 105-125 | DOI | MR | Zbl

[6] Dungey, N. Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z., Volume 247 (2004), pp. 765-794 | DOI | MR | Zbl

[7] Dungey, N. Some gradient estimates on covering manifolds, Bull. Pol. Acad. Sci. Math., Volume 52 (2004), pp. 437-443 | DOI | MR | Zbl

[8] Dungey, N. A note on time regularity for discrete time heat kernel, Semigroup Forum, Volume 72 (2006), pp. 404-410 | DOI | MR | Zbl

[9] Gromov, M. Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., Volume 53 (1981), pp. 53-73 | DOI | Numdam | MR | Zbl

[10] Hebisch, W.; Saloff-Coste, L. Gaussian estimates for Markov chains and random walks on groups, Ann. Probab., Volume 21 (1993), pp. 673-709 | DOI | MR | Zbl

[11] Ishiwata, S. Asymptotic behavior of a transition probability for a random walk on a nilpotent covering graph, Contemp. Math., Volume 347 (2004), pp. 57-68 | MR | Zbl

[12] Ishiwata, S. A Berry-Esseen type theorem on nilpotent covering graphs, Canad. J. Math., Volume 56 (2004), pp. 963-982 | DOI | MR | Zbl

[13] Russ, E. Riesz transform on graphs for p>2, unpublished manuscript

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