Duality of multiparameter Hardy spaces ${𝐇}^{𝐩}$ on spaces of homogeneous type
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 4, p. 645-685
In this paper, we introduce the Carleson measure space ${\mathrm{CMO}}^{p}$ on product spaces of homogeneous type in the sense of Coifman and Weiss [4], and prove that it is the dual space of the product Hardy space ${H}^{p}$ of two homogeneous spaces defined in [15]. Our results thus extend the duality theory of Chang and R. Fefferman [2,3] on ${H}^{1}\left({ℝ}_{+}^{2}×{ℝ}_{+}^{2}\right)$ with $\mathrm{BMO}\left({ℝ}_{+}^{2}×{ℝ}_{+}^{2}\right)$ which was established using bi-Hilbert transform. Our method is to use discrete Littlewood-Paley analysis in product spaces recently developed in [13] and [14].
Classification:  42B30,  42B35,  46B45
@article{ASNSP_2010_5_9_4_645_0,
author = {Han, Yongsheng and Li, Ji and Lu, Guozhen},
title = {Duality of multiparameter Hardy spaces $\mathbf{H^p}$ on spaces of homogeneous type},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {4},
year = {2010},
pages = {645-685},
zbl = {1213.42073},
mrnumber = {2789471},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_645_0}
}

Han, Yongsheng; Li, Ji; Lu, Guozhen. Duality of multiparameter Hardy spaces $\mathbf{H^p}$ on spaces of homogeneous type. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 9 (2010) no. 4, pp. 645-685. http://www.numdam.org/item/ASNSP_2010_5_9_4_645_0/

[1] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. LX/LXI (1990), 601–628. | MR 1096400 | Zbl 0758.42009

[2] S-Y. A. Chang and R. Fefferman, A continuous version of the duality of ${H}^{1}$ and BMO on the bi-disc, Ann. of Math. 112 (1980), 179–201. | MR 584078 | Zbl 0451.42014

[3] S-Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and ${H}^{p}$-theory on product domains, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 1–43. | Zbl 0557.42007

[4] R. R. Coifman and G. Weiss, “Analyse Harmonique Non-commutative sur Certains Espaces Homogeneous”, Lecture Notes in Math., Vol. 242, Springer-Verlag, Berlin, 1971. | MR 499948 | Zbl 0224.43006

[5] G. David, J. L. Journé and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985) 1–56. | MR 850408 | Zbl 0604.42014

[6] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. 126 (1987), 109–130. | MR 898053 | Zbl 0644.42017

[7] C. Fefferman and E. M. Stein, ${H}^{p}$ spaces of several variables, Acta Math. 129 (1972), 137–195. | MR 447953 | Zbl 0257.46078

[8] C. Fefferman and E. M. Stein, Some maximal inequalityies, Amer. J. Math. 93 (1971), 107–116. | MR 284802 | Zbl 0222.26019

[9] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces , J. Funct. Anal. 93(1990), 34–170. | MR 1070037 | Zbl 0716.46031

[10] S. H. Ferguson and M. Lacey, A characterization of product BMO by commutators, Acta Math. 189 (2002), 143–160. | MR 1961195 | Zbl 1039.47022

[11] G. Folland and E. M. Stein, “Hardy Spaces on Homogeneous Groups”, Princeton Univ. Press, Princeton, N. J., 1982. | MR 657581 | Zbl 0508.42025

[12] Y. Han, J. Li, G. Lu and P. Wang, ${H}^{p}\to {H}^{p}$ boundedness implies ${H}^{p}\to {L}^{p}$ boundedness, Forum Math., DOI 10-1515/FORM.2011.026. | MR 2820388 | Zbl 1225.42014

[13] Y. Han and G. Lu, Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals, preprint 2007 (available at: http://arxiv.org/abs/0801.1701).

[14] Y. Han and G. Lu, Endpoint estimates for singular integral operators and multi-parameter Hardy spaces associated with Zygmund dilation, to appear.

[15] Y. Han and G. Lu, Some recent works on multiparameter Hardy space theory and discrete Littlewood-Paley analysis, In: “Trends in Partial Differential Equations”, ALM 10, High Education Press and International Press (2009), Beijing-Boston, 99-191. | MR 2648281 | Zbl 1201.42016

[16] Y. Han, Discrete $Calder\stackrel{´}{o}n-type$ reproducing formula, Acta Math. Sin. (Engl.Ser.) 16 (2000), 277–294. | MR 1778708 | Zbl 0978.42010

[17] Y. Han, G. Lu and Y. Xiao, Discerete Littlewood-Paley analysis and multiparameter Hardy space theory on space of homogeneous type, preprint.

[18] Y. Han and E. Sawyer, Littlewood-paley theorem on space of homogeneous type and classical function spaces, Mem. Amer. Math. Soc. 110 (1994), 1–126. | MR 1214968 | Zbl 0806.42013

[19] J. L. Journé, Calderón-Zygmund operators on product space, Rev. Mat. Iberoamericana 1 (1985), 55–92. | MR 836284 | Zbl 0634.42015

[20] M. Lacey, S. Petermichl, J. Pipher and B. Wick, Multiparameter riesz commutators, Amer. J. Math. 131 (2009), 731–769. | MR 2530853 | Zbl 1170.42003

[21] M. Lacey and E. Terwilleger, Hankel operators in several complex variables and product BMO, Houston J. Math. 35 (2009), 159–183. | MR 2491875 | Zbl 1163.47024

[22] Y. Meyer, From wavelets to atoms, In: “150 Years of Mathematics at Washington University in St. Louis”, Gary Jensen and Steven Krantz (eds.), papers from the conference celebrating the sesquicentennial of mathematics held at Washington University, St. Louis, MO, October 3-5, 2003, 105–117. | Zbl 1100.42031

[23] R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (1979), 271–309. | MR 546296 | Zbl 0431.46019

[24] D. Muller, F. Ricci and E. M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg(-type) groups, I, Invent. Math. 119 (1995), 119–233. | MR 1312498 | Zbl 0857.43012

[25] A. Nagel, F. Ricci and E. M. Stein, Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal. 181 (2001), 29–118. | MR 1818111 | Zbl 0974.22007

[26] J. Pipher, Journe’s covering lemma and its extension to higher dimensions, Duke Math. J. 53 (1986), 683–690. | MR 860666 | Zbl 0645.42018

[27] F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), 637–670. | Numdam | MR 1182643 | Zbl 0760.42008

[28] E. Sawyer and R. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813–874. | MR 1175693 | Zbl 0783.42011

[29] E. M. Stein, “Singular Integral and Differentiability Properties of Functions”, Vol. 30, Princeton Univ. Press, Princeton, NJ, 1970. | MR 290095 | Zbl 0207.13501

[30] E. M. Stein, “Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals”, Princeton Univ. Press, Princeton, NJ, 1993. | MR 1232192 | Zbl 0821.42001