LAUSR

Abstract 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 K F ( 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 K F ( L ) implies continuity of F.