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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). 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 no two parameter group of 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 
$\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ that is, in a sense, intermediate between that of 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 Littlewood–Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.
