# Journal of Operator Theory

Volume 46, Issue 1, Summer 2001 pp. 45-61.

Higher order operators and gaussian bounds on Lie groups of polynomial growth**Authors**: Nick Dungey

**Author institution:**Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

**Summary:**Let $G$ be a connected Lie group of polynomial growth. We consider $m$-th order subelliptic differential operators $H$ on $G$, the semigroups $S_t = {\rm e}^{-t H}$ and the corresponding heat kernels $K_t$. For a large class of $H$ with $m \geq 4$ we demonstrate equivalence between the existence of Gaussian bounds on $K_t$, with ``good" large $t$ behaviour, and the existence of ``cutoff" functions on $G$. By results of [14], such cutoff functions exist if and only if $G$ is the local direct product of a compact Lie group and a nilpotent Lie group.

**Keywords:**Lie group, heat kernel, higher-order differential operators

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