Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-30T21:22:36.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Some regularity estimates for convolution semigroups on a group of polynomial growth

Part of: Partial differential equations Lie groups

Published online by Cambridge University Press:  09 April 2009

Nick Dungey
Nick Dungey
Affiliation:
School of Mathematics, University fo New South Wales, Sydney 2052, Australia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a convolution semigroup satisfying Gaussian estimates on a group G of polynomial volume growth. If Q is a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction of Q. Applications to heat kernels and convolution powers are given.

MSC classification

Secondary: 22E30: Analysis on real and complex Lie groups 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 2 , October 2004 , pp. 249 - 268
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Alexopoulos, G., ‘An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth’, Canad. J. Math. 44 (1992), 691–727.CrossRefGoogle Scholar
[2]Alexopoulos, G., ‘Random walks on discrete groups of polynomial volume growth’, Ann. Probab. 30 (2002), 723–801.CrossRefGoogle Scholar
[3]Alexopoulos, G., ‘Sub-Laplacians with drift on Lie groups of polynomial volume growth’, Mem. Amer. Math. Soc. 155 (2002), number 739.Google Scholar
[4]Bass, H., ‘The degree of polynomial growth of finitely generated nilpotent groups’, Proc. London Math. Soc. 25 (1972), 603–614.Google Scholar
[5]Baumslag, G., Lecture notes on nilpotent groups, CBMS Regional Conference Series in Mathematics No. 2 (Amer. Math. Soc., Providence, R.I., 1971).Google Scholar
[6]Butzer, P. L. and Berens, H., Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften Band 145 (Springer, New York, 1967).CrossRefGoogle Scholar
[7]Dungey, N., ‘High order regularity for subelliptic operators on Lie groups of polynomial growth’, Rev. Math. Iberoamericana, to appear.Google Scholar
[8]Dungey, N., ter Elst, A. F. M. and Robinson, D. W., Analysis on Lie groups with polynomial growth, Progress in Mathematics 214 (Birkhäuser, Boston, MA, 2003).Google Scholar
[9]Duong, X. T. and Robinson, D. W., ‘Semigroup kernels, Poisson bounds, and holomorphic functional calculus’, J. Funct. Anal. 142 (1996), 89–129.Google Scholar
[10]ter Elst, A. F. M., ‘Derivatives of kernels associated to complex subelliptic operators’, Bull. Austral. Math. Soc. 67 (2003), 393–406.Google Scholar
[11]ter Elst, A. F. M. and Robinson, D. W., ‘Weighted subcoercive operators on Lie groups’, J. Funct. Anal. 157 (1998), 88–163.CrossRefGoogle Scholar
[12]ter Elst, A. F. M. and Robinson, D. W., ‘Gaussian bounds for complex subelliptic operators on Lie groups of polynomial growth’, Bull. Austral. Math. Soc. 67 (2003), 201–218.CrossRefGoogle Scholar
[13]ter Elst, A. F. M., Robinson, D. W. and Sikora, A., ‘Riesz transforms and Lie groups of polynomial growth’, J. Funct. Anal. 162 (1999), 14–51.CrossRefGoogle Scholar
[14]Gromov, M., ‘Groups of polynomial growth and expanding maps’, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–78.CrossRefGoogle Scholar
[15]Hebisch, W. and Saloff-Coste, L., ‘Gaussian estimates for Markov chains and random walks on groups’, Ann. Probab. 21 (1993), 673–709.CrossRefGoogle Scholar
[16]Raghunathan, M. S., Discrete subgroups of Lie groups (Springer, New York, 1972).Google Scholar
[17]Robinson, D. W., Elliptic operators and Lie groups, Oxford Math. Monographs (Clarendon Press, Oxford, 1991).Google Scholar
[18]Saloff-Coste, L., ‘Analyse sur les groupes de Lie à croissance polynômiale’, Ark. Mat. 28 (1990), 315–331.Google Scholar
[19]Varopoulos, N. T., ‘Analysis on nilpotent groups’, J. Funct. Anal. 66 (1986), 406–431.Google Scholar
[20]Varopoulos, N. T., ‘Analysis on Lie groups’, J. Funct. Anal. 76 (1988), 346–410.CrossRefGoogle Scholar
[21]Varopoulos, N. T., Saloff-Coste, L. and Coulhon, T., Analysis and geometry on groups, Cambridge Tracts in Math. 100 (Cambridge University Press, Cambridge, 1992).Google Scholar