## Analysis & PDE

• Anal. PDE
• Volume 7, Number 7 (2014), 1465-1534.

### Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

#### Abstract

Marcinkiewicz multipliers are $Lp$ bounded for $1 on the Heisenberg group $ℍn≃ℂn×ℝ$, 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 $ℂn×ℝ$, while there is no two-parameter group of automorphic dilations on $ℍn$. 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, that is in a sense “intermediate” between the classical Hardy spaces $Hp$ and the product Hardy spaces $Hproductp$ on $ℂn×ℝ$ 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. 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 $L2$.

#### Article information

Source
Anal. PDE, Volume 7, Number 7 (2014), 1465-1534.

Dates
Revised: 30 January 2014
Accepted: 1 April 2014
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731603

Digital Object Identifier
doi:10.2140/apde.2014.7.1465

Mathematical Reviews number (MathSciNet)
MR3293443

Zentralblatt MATH identifier
1318.42026

#### Citation

Han, Yongsheng; Lu, Guozhen; Sawyer, Eric. Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group. Anal. PDE 7 (2014), no. 7, 1465--1534. doi:10.2140/apde.2014.7.1465. https://projecteuclid.org/euclid.apde/1513731603

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