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

top
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

top

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

top
  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 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.