Duality of multiparameter Hardy spaces ${𝐇}^{𝐩}$ on spaces of homogeneous type

Han, Yongsheng; Li, Ji; Lu, Guozhen

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 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 $H^1({ℝ}_{+}^2\times {ℝ}_{+}^2)$ with $\mathrm{BMO}({ℝ}_{+}^2\times {ℝ}_{+}^2)$ 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].

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

[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 | Zbl

[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

[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 | Zbl

[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. | EuDML | MR | Zbl

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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

[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 | Zbl

[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. | EuDML | MR | Zbl

[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 | Zbl

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

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

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

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

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