Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings

Satoshi Ishiwata[1]

  • [1] Institute of Mathematics University of Tsukuba 1-1-1 Tennoudai, 305-8571 Ibaraki JAPAN

Annales mathématiques Blaise Pascal (2007)

  • Volume: 14, Issue: 1, page 93-102
  • ISSN: 1259-1734

Abstract

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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.

How to cite

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Ishiwata, Satoshi. "Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings." Annales mathématiques Blaise Pascal 14.1 (2007): 93-102. <http://eudml.org/doc/10543>.

@article{Ishiwata2007,
abstract = {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.},
affiliation = {Institute of Mathematics University of Tsukuba 1-1-1 Tennoudai, 305-8571 Ibaraki JAPAN},
author = {Ishiwata, Satoshi},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Gradient estimates; Random walks; Gaussian estimates for the heat kernel; gradient estimates; random walks},
language = {eng},
month = {1},
number = {1},
pages = {93-102},
publisher = {Annales mathématiques Blaise Pascal},
title = {Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings},
url = {http://eudml.org/doc/10543},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Ishiwata, Satoshi
TI - Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings
JO - Annales mathématiques Blaise Pascal
DA - 2007/1//
PB - Annales mathématiques Blaise Pascal
VL - 14
IS - 1
SP - 93
EP - 102
AB - 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.
LA - eng
KW - Gradient estimates; Random walks; Gaussian estimates for the heat kernel; gradient estimates; random walks
UR - http://eudml.org/doc/10543
ER -

References

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  1. P. Auscher, T. Coulhon, X. T. Duong, S. Hofmann, Riesz transform on manifolds and heat kernel regurality, Ann. Scient. Éc. Norm. Sup. 37 (2004), 911-957 Zbl1086.58013MR2119242
  2. S. Blunck, Perturbation of analytic operators and temporal regularity, Colloq. Math. 86 (2000), 189-201 Zbl0961.47005MR1808675
  3. M. Christ, 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. Special Issue (1995), 141-151 Zbl0889.60007MR1364882
  4. T. Coulhon, X. T. Duong, Riesz transforms for 1 ≤ p ≤ 2 , Trans. Amer. Math. Soc. 351 (1999), 1151-1169 Zbl0973.58018MR1458299
  5. E. B. Davies, Non-gaussian aspects of heat kernel behaviour, J. London Math. Soc. 55 (1997), 105-125 Zbl0879.35064MR1423289
  6. N. Dungey, Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z. 247 (2004), 765-794 Zbl1080.58022MR2077420
  7. N. Dungey, Some gradient estimates on covering manifolds, Bull. Pol. Acad. Sci. Math. 52 (2004), 437-443 Zbl1112.58027MR2128280
  8. N. Dungey, A note on time regularity for discrete time heat kernel, Semigroup Forum 72 (2006), 404-410 Zbl1102.47016MR2228535
  9. M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73 Zbl0474.20018MR623534
  10. W. Hebisch, L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993), 673-709 Zbl0776.60086MR1217561
  11. S. Ishiwata, Asymptotic behavior of a transition probability for a random walk on a nilpotent covering graph, Contemp. Math. 347 (2004), 57-68 Zbl1061.22009MR2077030
  12. S. Ishiwata, A Berry-Esseen type theorem on nilpotent covering graphs, Canad. J. Math. 56 (2004), 963-982 Zbl1062.22018MR2085630
  13. E. Russ, Riesz transform on graphs for p &gt; 2 , unpublished manuscript 

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