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.

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