Spectral Multipliers on $2$-Step Stratified Groups, II

Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $$\mathcal{L}$_1,\dots, $\mathcal{L}$_n$on$G$, one of which is Rockland, we study the convolution operators $m({\mathcal{L}}_1,\dots ,{\mathcal{L}}_n)$ 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 ${\mathcal{L}}_1,\dots ,{\mathcal{L}}_n$ 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 $m({\mathcal{L}}_1,\dots ,{\mathcal{L}}_n)$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\mathrm{\infty }$ (`Riemann-Lebesgue lemma’); ii) if the convolution kernel of the operator $m({\mathcal{L}}_1,\dots ,{\mathcal{L}}_n)$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.