Spectral multiplier theorems of Euclidean type on new classes of 2-step stratified groups

Abstract

From a theorem of Christ and Mauceri and Meda it follows that, for a homogeneous sublaplacian $L$ on a $2$-step stratified group $G$ with Lie algebra $\mathfrak{g}$, an operator of the form $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for $1<p<\infty$ if the spectral multiplier $F$ satisfies a scale-invariant smoothness condition of order $s>Q/2$, where $Q=\dim \mathfrak{g}+\dim [\mathfrak{g},\mathfrak{g}]$ is the homogeneous dimension of $G$. Here we show that the condition can be pushed down to $s>d/2$, where $d=\dim \mathfrak{g}$ is the topological dimension of $G$, provided that $d\le 7$ or $\dim [\mathfrak{g},\mathfrak{g}]\le 2$.

33 pages

References in corpus (1)

• $L^p$ spectral multipliers on the free group $N_{3,2}$

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