Spectral multipliers on $2$-step groups: topological versus homogeneous dimension

Abstract

Let $G$ be a $2$-step stratified group of topological dimension $d$ and homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a theorem due to Christ and to Mauceri and 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 $s>Q/2$. It is known that, for several $2$-step groups and sub-Laplacians, the threshold $Q/2$ in the smoothness condition is not sharp and in many cases it is possible to push it down to $d/2$. Here we show that, for all $2$-step groups and sub-Laplacians, the sharp threshold is strictly less than $Q/2$, but not less than $d/2$.

17 pages

References in corpus (3)

• Spectral multiplier theorems of Euclidean type on new classes of 2-step stratified groups
• $L^p$ spectral multipliers on the free group $N_{3,2}$
• Spectral multipliers on Heisenberg-Reiter and related groups

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