Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

Abstract

Marcinkiewicz multipliers are $L^p$ bounded for $1<p<\infty$ on the Heisenberg group ${ℍ}^n\simeq {ℂ}^n\times ℝ$, 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 ${ℂ}^n\times ℝ$, while there is no two-parameter group of automorphic dilations on ${ℍ}^n$. This lack of automorphic dilations underlies the failure of such multipliers to be in general bounded on the classical Hardy space $H^1$ 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 $H_{flag}^p$ on the Heisenberg group, $0<p\le 1$, that is in a sense “intermediate” between the classical Hardy spaces $H^p$ and the product Hardy spaces $H_{product}^p$ on ${ℂ}^n\times ℝ$ developed by A. Chang and R. Fefferman. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on $H_{flag}^p$, as well as from $H_{flag}^p$ to $L^p$, for $0<p\le 1$. We also characterize the dual spaces of $H_{flag}^1$ and $H_{flag}^p$, and establish a Calderón–Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces $H_{flag}^p$. In particular, this recovers some $L^p$ results of Müller, Ricci, and Stein (but not their sharp versions) by interpolating between those for $H_{flag}^p$ and $L^2$.