Spectral multipliers on Heisenberg-Reiter and related groups

Abstract

Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$ for $1<p<\infty$ if $F$ satisfies a scale-invariant smoothness condition of order $s>Q/2$. Under suitable assumptions on $G$ and $L$, here we show that a smoothness condition of order $s>d/2$ is sufficient. This extends to a larger class of 2-step groups the results for the Heisenberg and related groups by Müller and Stein and by Hebisch, and for the free group $N_{3,2}$ by Müller and the author.

References in corpus (1)

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

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• Bilinear Bochner-Riesz Means on Métivier groups
• An $L^p$-spectral multiplier theorem with sharp $p$-specific regularity bound on Heisenberg type groups
• A sharp multiplier theorem for solvable extensions of Heisenberg and related groups