Spectral multipliers on 2-step stratified groups, II

Given a graded group G and commuting, formally self-adjoint, left-invariant, homogeneous differential operators L1, … , Ln on G, one of which is Rockland, we study the convolution operators m(L1, … , Ln) 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 L1, … , Ln 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(L1, … , Ln) 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(L1, … , Ln) is a Schwartz function, then m equals almost everywhere a Schwartz function.