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, greater than or equal to 4 we demonstrate equivalence between the existence of Gau ssian bounds on K-t, with "good" large t behaviour, and the existence of "c utoff" functions on G. By results of [14], such cutoff functions exist if a nd only if G is the local direct product of a compact Lie group and a nilpo tent Lie group.