Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups

Abstract

The Marcinkiewicz multipliers are $L^p$ bounded for $1<p<\infty$ on the Heisenberg group ${\mathbb{H}}^n\simeq {\mathbb{C}}^n\times \mathbb{R}$ (Müller, Ricci and Stein \cite{MRS}). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on ${\mathbb{C}}^n\times \mathbb{R}$, while there is \emph{no} two parameter group of \emph{automorphic} dilations on ${\mathbb{H}}^n$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group ${\mathbb{H}}^n\simeq {\mathbb{C}}^n\times \mathbb{R}$ in the sense `intermediate' between the classical Lipschitz space on the Heisenberg group ${\mathbb{H}}^n$ and the product Lipschitz space on ${\mathbb{C}}^n\times \mathbb{R}$. We characterize this flag Lipschitz space via the Littelewood-Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.