Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

Han, Yongsheng; Lu, Guozhen; Sawyer, Eric

doi:10.2140/apde.2014.7.1465

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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

DOI: 10.2140/apde.2014.7.1465

Abstract

Marcinkiewicz multipliers are $L^{p}$ bounded for $1 < p < \infty$ on the Heisenberg group $ℍ^{n} ≃ ℂ^{n} \times ℝ$ , 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, $0 < p \leq 1$ , 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 $0 < p \leq 1$ . 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 $L^{2}$ .

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

Milestones

Received: 24 January 2013

Revised: 30 January 2014

Accepted: 1 April 2014

Published: 12 December 2014

Authors

Yongsheng Han
Department of Mathematics Auburn University Auburn, AL 36849 United States

Guozhen Lu
Department of Mathematics Wayne State University 656 W. Kirby Street Detroit, MI 48202 United States

Eric Sawyer
Department of Mathematics and Statistics McMaster University 1280 Main Street West Hamilton, ON L8S 4K1 Canada

Vol. 7, No. 7, 2014