Spectral Multipliers on 2-Step Stratified Groups, I

Given a 2-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian L and a family T of elements of the derived algebra, we study the convolution kernels associated with the operators of the form m(L,- iT). Under suitable conditions, we prove that: (i) if the convolution kernel of the operatorm(L,- iT) belongs to L1, thenm equals almost everywhere a continuous function vanishing at 8 Riemann-Lebesgue lemma'); (ii) if the convolution kernel of the operator m(L,- iT) is a Schwartz function, then m equals almost everywhere a Schwartz function.