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 operator m(L,-iT) belongs to L1, then m equals almost everywhere a continuous function vanishing at ∞ (‘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.