Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups
  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. 
Yanchang Han, Yongsheng Han, Ji Li and Chaoqiang Tan
math.FA