Title: Spectral multipliers on 2-step groups: topological versus homogeneous dimension
Abstract: Let $G$ be a stratified group of topological dimension $d$ and homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a theorem of M. Christ, independently proven also by G. Mauceri and S. Meda, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. However, in particular for stratified groups of step 2, several classes of such groups have been identified by now for which the threshold $Q/2$ in the smoothness condition is not sharp, and indeed can be pushed down to $d/2$.
In my talk, I shall present the state of the art in this field and then concentrate on more recent joint work with Alessio Martini, in which we have shown that, for all $2$-step groups, the sharp threshold is indeed always strictly less than $Q/2$, but not less than $d/2$.