Vol. 7, No. 7, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 Lp bounded for 1 < p < on the Heisenberg group n n × , 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 n × , while there is no two-parameter group of automorphic dilations on n. This lack of automorphic dilations underlies the failure of such multipliers to be in general bounded on the classical Hardy space H1 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 Hflagp on the Heisenberg group, 0 < p 1, that is in a sense “intermediate” between the classical Hardy spaces Hp and the product Hardy spaces Hproductp on n × developed by A. Chang and R. Fefferman. We show that flag singular integral operators, which include the aforementioned Marcinkiewicz multipliers, are bounded on Hflagp, as well as from Hflagp to Lp, for 0 < p 1. We also characterize the dual spaces of Hflag1 and Hflagp, and establish a Calderón–Zygmund decomposition that yields standard interpolation theorems for the flag Hardy spaces Hflagp. In particular, this recovers some Lp results of Müller, Ricci, and Stein (but not their sharp versions) by interpolating between those for Hflagp and L2.

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