Heat kernel and Green function estimates on noncompact symmetric spaces. II.

*(English)*Zbl 0988.22006
Taylor, J. C. (ed.), Topics in probability and Lie groups: boundary theory. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes. 28, 1-9 (2001).

Let \(X=G/K\) be a noncompact symmetric space with heat kernel \(h_t\). Here \(h_t\) is the heat kernel associated to the scalar Laplace-Beltrami operator and based at the identity coset. In Part I of this paper [Geom. Funct. Anal. 9, 1035-1091 (1999; Zbl 0942.43005)], the authors proved optimal upper and lower bounds on \(h_t\) in the restricted region that the space variable is bounded by an arbitrarily large multiple of the time variable. The most difficult part of their proof concerned the behavior of spherical functions near the walls of the Weyl chamber. For this analysis, the authors made use of the asymptotic expansion of Trombi and Varadarajan for spherical functions along the walls. The present article shows that the proof of the bounds may be greatly simplified by using the fact that the heat kernel satisfies a parabolic Harnack inequality, thereby allowing the authors to avoid the complicated asymptotic expansion for spherical functions.

For the entire collection see [Zbl 0970.00015].

For the entire collection see [Zbl 0970.00015].

Reviewer: Jeffrey J.Mitchell (Waco)

##### MSC:

22E30 | Analysis on real and complex Lie groups |

22E46 | Semisimple Lie groups and their representations |

31C12 | Potential theory on Riemannian manifolds and other spaces |

43A80 | Analysis on other specific Lie groups |

43A85 | Harmonic analysis on homogeneous spaces |

43A90 | Harmonic analysis and spherical functions |