Let L be a sub-Laplacian on a connected Lie group G of
polynomial growth. It is well known that, if F:R→C is
in the Schwartz class S(R), then the convolution kernel
KF(L) of the operator F(L) is in the
Schwartz class S(G). Here we prove a sort of converse implication
for a class of groups G including all solvable noncompact groups of
polynomial growth. We also discuss the problem whether integrability of
KF(L) implies continuity of F.Comment: 27 page