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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].

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