Spectral Multipliers on 2-Step Stratified Groups, II

Given a graded group G and commuting, formally self-adjoint, left-invariant, homogeneous differential operators L_1,..., L_n on G, one of which is Rockland, we study the convolution operators m(L_1,..., L_n) and their convolution kernels, with particular reference to the case in which G is abelian and n=1, and the case in which G is a 2-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and L_1,..., L_n are either sub-Laplacians or central elements of the Lie algebra of G. Under suitable conditions, we prove that: i) if the convolution kernel of the operator m(L_1,..., L_n) belongs to L^1, then m equals almost everywhere a continuous function vanishing at infinity (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator m(L_1,..., L_n) is a Schwartz function, then m equals almost everywhere a Schwartz function.