Marcinkiewicz multipliers are bounded for on the Heisenberg group , as shown by D. Müller, F. Ricci, and E. M. Stein. This is surprising in that these multipliers are invariant under a two-parameter group of dilations on , while there is no two-parameter group of automorphic dilations on . This lack of automorphic dilations underlies the failure of such multipliers to be in general bounded on the classical Hardy space on the Heisenberg group, and also precludes a pure product Hardy space theory.
We address this deficiency by developing a theory of flag Hardy spaces on the Heisenberg group, , that is in a sense “intermediate” between the classical Hardy spaces and the product Hardy spaces on developed by A. Chang and R. Fefferman. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on , as well as from to , for . We also characterize the dual spaces of and , and establish a Calderón–Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces . In particular, this recovers some results of Müller, Ricci, and Stein (but not their sharp versions) by interpolating between those for and .
"Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group." Anal. PDE 7 (7) 1465 - 1534, 2014. https://doi.org/10.2140/apde.2014.7.1465