Analysis & PDE

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

Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group

Yongsheng Han, Guozhen Lu, and Eric Sawyer

Full-text: Open access

Abstract

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.

Article information

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

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

Permanent link to this document
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

Subjects
Primary: 42B15: Multipliers 42B35: Function spaces arising in harmonic analysis

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

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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