Open Access
Translator Disclaimer
2014 Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group
Yongsheng Han, Guozhen Lu, Eric Sawyer
Anal. PDE 7(7): 1465-1534 (2014). DOI: 10.2140/apde.2014.7.1465


Marcinkiewicz multipliers are Lp bounded for 1<p< on the Heisenberg group nn×, 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<p1, 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<p1. 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.


Download Citation

Yongsheng Han. Guozhen Lu. Eric Sawyer. "Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group." Anal. PDE 7 (7) 1465 - 1534, 2014.


Received: 24 January 2013; Revised: 30 January 2014; Accepted: 1 April 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1318.42026
MathSciNet: MR3293443
Digital Object Identifier: 10.2140/apde.2014.7.1465

Primary: 42B15 , 42B35

Keywords: Calderón reproducing formulas , discrete Calderón reproducing formulas , discrete Littlewood–Paley analysis , flag Hardy spaces , flag singular integrals

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 7 • 2014
Back to Top