# Journal of Operator Theory

Volume 42, Issue 2, Fall 1999 pp. 269-304.

Reduced heat kernels on homogeneous spaces**Authors**: A.F.M. ter Elst (1), and C.M.P.A. Smulders (2)

**Author institution:**(1) Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

(2) Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

**Summary:**If $S$ is the semigroup generated by an $n$-th order strongly elliptic operator on $L_p(X;{\rm d}x)$ associated with the left regular representation of a unimodular Lie group $G$ in the homogeneous space $X = G / M$, where $M$ is a compact subgroup of $G$, and $\kappa$ is the reduced heat kernel of $S$ defined by $$ (S_t \varphi)(x) = \int\limits_X \kappa_t(x\,;y) \, \varphi(y) \, {\rm d}y $$ then we prove Gaussian upper bounds for $\kappa_t$ and all its derivatives. For reduced heat kernels associated with irreducible unitary representations on nilpotent Lie groups we prove similar Gaussian bounds.

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