Flag Hardy Spaces and Marcinkiewicz Multipliers on the Heisenberg Group: an Expanded Version

Marcinkiewicz multipliers are L bounded for 1 < p < ∞ on the Heisenberg group H ≃ C × R (D. Muller, F. Ricci and E. M. Stein [25], [26]). This is surprising in that this class of multipliers is invariant under a two parameter group of dilations on C × R, while there is no two parameter group of automorphic dilations on H. This lack of automorphic dilations underlies the inability of classical one or two parameter Hardy space theory to handle Marcinkiewicz multipliers on H when 0 < p ≤ 1. We address this deficiency by developing a theory of flag Hardy spaces H p flag on the Heisenberg group, 0 < p ≤ 1, that is in a sense ‘intermediate’ between the classical Hardy spaces H and the product Hardy spaces H product on C×R (A. Chang and R. Fefferman ([3], [4], [8], [9], [10]). We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on H flag , as well as from H flag to L, for 0 < p ≤ 1. We also characterize the dual spaces of H flag and H flag , and establish a Calderón-Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces H flag . In particular, this recovers the L results in [25] (but not the sharp results in [26]) by interpolating between those for H flag and L.