Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T07:03:32.935Z Has data issue: false hasContentIssue false
  • English
  • Français

Heat Kernels on homogeneous spaces

Part of: Locally compact groups and their algebras Lie groups Abstract harmonic analysis Parabolic equations and systems

Published online by Cambridge University Press:  09 April 2009

C. M. P. A. Smulders
C. M. P. A. Smulders
Affiliation:
Department of Mathematics and Comp. Sci., Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 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.

Let a1… ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S: X × G → C is a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in Lp (X;dx) for all p ε [1,].

Let Ai be the infinitesimal generator with respect to U and the direction ai, for all i ∈ { 1… d}. We consider n–th order strongly elliptic operators H = ΣcαAα with complex coefficients cα. We show that the semigroup S generated by the closure of H has a reduced heat kernel K and we derive upper bounds for k and all its derivatives.

Keywords

reduced heat kernelhomogeneous spacequasi-invariant measureGaussian estimatestrongly elliptic operatorcocycle representation

MSC classification

Secondary: 43A85: Analysis on homogeneous spaces 22D30: Induced representations 22E25: Nilpotent and solvable Lie groups 22E45: Representations of Lie and linear algebraic groups over real fields 35K05: Heat equation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 78 , Issue 1 , February 2005 , pp. 109 - 147
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Arendt, W. and Bukhvalov, A. V., ‘Integral representations of resolvents and semigroups’, Forum Math. 6 (1994), 111–135.CrossRefGoogle Scholar
[2]Aronson, D. G., ‘Bounds for the fundamental solution of a parabolic equation’, Bull. Amer. Math. Soc. 73 (1967), 890–896.CrossRefGoogle Scholar
[3]Davies, E. B., Heat kernels and spectral theory, Cambridge Tracts in Mathematics 92 (Cambridge University Press, Cambridge, 1989).CrossRefGoogle Scholar
[4]ter Elst, A. F. M. and Robinson, D. W., ‘Subcoercive and subelliptic operators on Lie groups: variable coefficients’, Publ. RIMS. Kyoto Univ. 29 (1993), 745–801.CrossRefGoogle Scholar
[5]ter Elst, A. F. M., and Robinson, D. W., ‘Weighted subcoercive operators on Lie groups’, J. Funct. Anal. 157 (1998), 88–163.CrossRefGoogle Scholar
[6]ter Elst, A. F. M. and Smulders, C. M. P. A., ‘Reduced heat kernels on homogeneous spaces’, J. Operator Theory 42 (1999), 269–304.Google Scholar
[7]Grigor'yan, A., ‘The heat equation on noncompact Riemannian manifolds’, Math. USSR Sbornik 72 (1992), 47–77.CrossRefGoogle Scholar
[8]Grigor'yan, A., ‘Heat kernel upper bounds on a complete non-compact manifold’, Rev. Mat. Iberoamericana 10 (1994), 395–452.CrossRefGoogle Scholar
[9]Jerison, D. S. and Sánchez-Calle, A., ‘Estimates for the heat kernel for a sum of squares of vector fields’, Ind. Univ. Math. J. 35 (1986), 835–854.CrossRefGoogle Scholar
[10]Koornwinder, T. H., Representations of locally compact groups with applications Part II, MC Syllabus 38.2 (Mathematical Centre, Amsterdam, 1979).Google Scholar
[11]Li, P. and Yau, S. T., ‘On the upper estimate of the heat kernel of a complete Riemannian manifold’, Amer. J. Math. 103 (1981), 1021–1063.Google Scholar
[12]Lohoué, N. and Mustapha, S., ‘Sur les transformées de riesz sur les groupes de Lie moyennables et sur certains espaces homogènes’, Canad. J. Math. 50 (1998), 1090–1104.CrossRefGoogle Scholar
[13]Maheux, P., Analyse et géometric sur les espaces homogènes (Ph.D. Thesis, Université de Paris VI, Paris, 1991).Google Scholar
[14]Maheux, P., ‘Estimations du noyau de la chaleur sur les espaces homogènes’, J. Geom. Anal. 8 (1998), 65–96.CrossRefGoogle Scholar
[15]Robinson, D. W., Elliptic operators and Lie groups, Oxford Mathematical Monographs (Oxford University Press, Oxford, 1991).CrossRefGoogle Scholar
[16]Smulders, C. M. P. A., Reduced heat kernels on homogeneous spaces (Ph.D. Thesis, Eindhoven University of Technology, Eindhoven, the Netherlands, 2000).Google Scholar
[17]Triebel, H., Interpolation theory, function spaces, differential operators (North-Holland, Amsterdam, 1978).Google Scholar
[18]Varopoulos, N. T., ‘The heat kernel on Lie groups’, Rev. Math. Iberoamericana 12 (1996), 147–186.Google Scholar