#### Vol. 7, No. 7, 2014

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Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

### Yongsheng Han, Guozhen Lu and Eric Sawyer

Vol. 7 (2014), No. 7, 1465–1534
##### Abstract

Marcinkiewicz multipliers are ${L}^{p}$ bounded for $1 on the Heisenberg group ${ℍ}^{n}\simeq {ℂ}^{n}×ℝ$, 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 ${ℂ}^{n}×ℝ$, 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, that is in a sense “intermediate” between the classical Hardy spaces ${H}^{p}$ and the product Hardy spaces ${H}_{product}^{p}$ on ${ℂ}^{n}×ℝ$ 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. 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 ${L}^{2}$.

##### Keywords
flag singular integrals, flag Hardy spaces, Calderón reproducing formulas, discrete Calderón reproducing formulas, discrete Littlewood–Paley analysis
##### Mathematical Subject Classification 2010
Primary: 42B15, 42B35