Higher order operators and Gaussian bounds on Lie groups of polynomial growth

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 = e^-tH and the corresponding heat kernels K_t. For a large class of H with m ≥ 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.