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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • L. Carleson, “A counterexample for measures bounded on $H^p$ for the bidisc”, technical report, Institut Mittag-Leffler, 1974.
  • S.-Y. A. Chang, “Carleson measure on the bi-disc”, Ann. of Math. $(2)$ 109:3 (1979), 613–620.
  • S.-Y. A. Chang and R. Fefferman, “A continuous version of duality of $H\sp{1}$ with BMO on the bidisc”, Ann. of Math. $(2)$ 112:1 (1980), 179–201.
  • S.-Y. A. Chang and R. Fefferman, “The Calderón–Zygmund decomposition on product domains”, Amer. J. Math. 104:3 (1982), 455–468.
  • S.-Y. A. Chang and R. Fefferman, “Some recent developments in Fourier analysis and $H\sp p$-theory on product domains”, Bull. Amer. Math. Soc. $($N.S.$)$ 12:1 (1985), 1–43.
  • R. R. Coifman and G. Weiss, Transference methods in analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 31, American Mathematical Society, Providence, 1976.
  • R. Fefferman, “Multiparameter Fourier analysis”, pp. 47–130 in Beijing lectures in harmonic analysis (Beijing, 1984), edited by E. M. Stein, Ann. of Math. Stud. 112, Princeton Univ. Press, Princeton, 1986.
  • R. Fefferman, “Harmonic analysis on product spaces”, Ann. of Math. $(2)$ 126:1 (1987), 109–130. 90e:42030
  • R. A. Fefferman, “Multiparameter Calderón–Zygmund theory”, pp. 207–221 in Harmonic analysis and partial differential equations (Chicago, 1996), edited by M. Christ et al., Univ. Chicago Press, Chicago, 1999.
  • C. Fefferman and E. M. Stein, “$H\sp{p}$ spaces of several variables”, Acta Math. 129:3-4 (1972), 137–193.
  • R. Fefferman and E. M. Stein, “Singular integrals on product spaces”, Adv. in Math. 45:2 (1982), 117–143.
  • S. H. Ferguson and M. T. Lacey, “A characterization of product BMO by commutators”, Acta Math. 189:2 (2002), 143–160.
  • M. Frazier and B. Jawerth, “A discrete transform and decompositions of distribution spaces”, J. Funct. Anal. 93:1 (1990), 34–170.
  • D. Geller and A. Mayeli, “Continuous wavelets and frames on stratified Lie groups, I”, J. Fourier Anal. Appl. 12:5 (2006), 543–579.
  • J. Gilbert, Y. Han, J. Hogan, J. Lakey, D. Weiland, and G. Weiss, “Smooth molecular decompositions of functions and singular integral operators”, Mem. Am. Math. Soc. 742 (2002), 74.
  • R. F. Gundy and E. M. Stein, “$H\sp{p}$ theory for the poly-disc”, Proc. Nat. Acad. Sci. U.S.A. 76:3 (1979), 1026–1029.
  • Y. S. Han, “Calderón-type reproducing formula and the $Tb$ theorem”, Rev. Mat. Iberoamericana 10:1 (1994), 51–91.
  • Y.-S. Han, “Plancherel–Pólya type inequality on spaces of homogeneous type and its applications”, Proc. Amer. Math. Soc. 126:11 (1998), 3315–3327.
  • Y. Han and G. Lu, “Discrete Littlewood–Paley–Stein theory and multi-parameter Hardy spaces associated with flag singular integrals”, preprint, 2008.
  • Y. Han and G. Lu, “Some recent works on multiparameter Hardy space theory and discrete Littlewood–Paley analysis”, pp. 99–191 in Trends in partial differential equations, Adv. Lect. Math. (ALM) 10, International Press, 2010.
  • Y. Han, G. Lu, and E. Sawyer, “Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group: an expanded version”, preprint, 2012.
  • Y. Han, J. Li, and G. Lu, “Multiparameter Hardy space theory on Carnot–Carathéodory spaces and product spaces of homogeneous type”, Trans. Amer. Math. Soc. 365:1 (2013), 319–360.
  • Y. Han, C. Lin, G. Lu, Z. Ruan, and E. T. Sawyer, “Hardy spaces associated with different homogeneities and boundedness of composition operators”, Rev. Mat. Iberoam. 29:4 (2013), 1127–1157.
  • B. Jessen, J. Marcinkiewicz, and A. Zygmund, “Note on the differentiability of multiple integrals”, Fundam. Math. 25 (1935), 217–234.
  • J.-L. Journé, “Calderón–Zygmund operators on product spaces”, Rev. Mat. Iberoamericana 1:3 (1985), 55–91.
  • J.-L. Journé, “A covering lemma for product spaces”, Proc. Amer. Math. Soc. 96:4 (1986), 593–598.
  • J.-L. Journé, “Two problems of Calderón–Zygmund theory on product-spaces”, Ann. Inst. Fourier $($Grenoble$)$ 38:1 (1988), 111–132.
  • M. Lacey and E. Terwilliger, “Little Hankel operators and product BMO”, lecture presented at harmonic analysis and its applications, 2005,
  • D. Müller, F. Ricci, and E. M. Stein, “Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, I”, Invent. Math. 119:2 (1995), 199–233.
  • D. Müller, F. Ricci, and E. M. Stein, “Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, II”, Math. Z. 221:2 (1996), 267–291.
  • A. Nagel and E. M. Stein, “On the product theory of singular integrals”, Rev. Mat. Iberoamericana 20:2 (2004), 531–561.
  • A. Nagel, F. Ricci, and E. M. Stein, “Singular integrals with flag kernels and analysis on quadratic CR manifolds”, J. Funct. Anal. 181:1 (2001), 29–118.
  • A. Nagel, F. Ricci, E. Stein, and S. Wainger, “Singular integrals with flag kernels on homogeneous groups, I”, Rev. Mat. Iberoam. 28:3 (2012), 631–722.
  • J. Pipher, “Journé's covering lemma and its extension to higher dimensions”, Duke Math. J. 53:3 (1986), 683–690.
  • M. Plancherel and G. Pólya, “Fonctieres entières et intégrales de Fourier multiples”, Comment. Math. Helv. 9 (1937), 224–248.
  • G. Pólya, “Fonctions entières et intégrales de Fourier multiples”, Comment. Math. Helv. 9:1 (1936), 224–248.
  • R. S. Strichartz, “Self-similarity on nilpotent Lie groups”, pp. 123–157 in Geometric analysis (Philadelphia, 1991), edited by E. L. Grinberg, Contemp. Math. 140, Amer. Math. Soc., Providence, 1992.
  • R. H. Torres, “Boundedness results for operators with singular kernels on distribution spaces”, Mem. Amer. Math. Soc. 90:442 (1991), viii+172.
  • J. T. Tyson, “Global conformal Assouad dimension in the Heisenberg group”, Conform. Geom. Dyn. 12 (2008), 32–57.