Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth

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 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.