Marcinkiewicz Multipliers and Lipschitz Spaces on Heisenberg Groups

The Marcinkiewicz multipliers are L-p bounded for 1 < p < infinity on the Heisenberg group H-n similar or equal to C-n x R (Muller, Ricci and Stein). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on C-n x R, while there is no two parameter group of automorphic dilations on H-n. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group H-n similar or equal to C-n x R in the sense, intermediate between the classical Lipschitz space on the Heisenberg group H-n and the product Lipschitz space on C-n x 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.