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Heat Kernels on homogeneous spaces

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

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: camsmul@cs.com
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Abstract

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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

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