Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups

Abstract

The Marcinkiewicz multipliers are $L^{p}$ bounded for $1<p<\infty $ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (M\"uller, 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.