Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups

The Marcinkiewicz multipliers are Lp bounded for 1 < p < ∞ on the Heisenberg group Hⁿ Cⁿ × R (Müller, Ricci, and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on Cⁿ × R, while there is no two parameter group of automorphic dilations on Hⁿ. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group Hⁿ Cⁿ × R that is, in a sense, intermediate between that of the classical Lipschitz space on the Heisenberg group Hⁿ and the product Lipschitz space on Cⁿ × 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.