Various characterizations of product Hardy space

Li, Ji; Song, Liang; Tan, Chaoqiang

doi:10.1090/S0002-9939-2011-10852-9

Various characterizations of product Hardy space
HTML articles powered by AMS MathViewer

by Ji Li, Liang Song and Chaoqiang Tan

Proc. Amer. Math. Soc. 139 (2011), 4385-4400

DOI: https://doi.org/10.1090/S0002-9939-2011-10852-9

Published electronically: April 22, 2011

PDF | Request permission

Abstract:

This article deals with the characterizations of Hardy space $H^1$ on $\mathbb {R}^n\times \mathbb {R}^m$ using different norms on distinct variables. This result can be applied to the boundedness of certain operators on $H^1(\mathbb {R}^n\times \mathbb {R}^m)$.

References

Sun-Yung A. Chang and Robert Fefferman, The Calderón-Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), no. 3, 455–468. MR 658542, DOI 10.2307/2374150
Sun-Yung A. Chang and Robert Fefferman, Some recent developments in Fourier analysis and $H^p$-theory on product domains, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 1–43. MR 766959, DOI 10.1090/S0273-0979-1985-15291-7
R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
Robert Fefferman, Calderón-Zygmund theory for product domains: $H^p$ spaces, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 4, 840–843. MR 828217, DOI 10.1073/pnas.83.4.840
Robert Fefferman, Multiparameter Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 47–130. MR 864371
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
R. F. Gundy and E. M. Stein, $H^{p}$ theory for the poly-disc, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1026–1029. MR 524328, DOI 10.1073/pnas.76.3.1026
Yongsheng Han, Ming-Yi Lee, Chin-Cheng Lin, and Ying-Chieh Lin, Calderón-Zygmund operators on product Hardy spaces, J. Funct. Anal. 258 (2010), no. 8, 2834–2861. MR 2593346, DOI 10.1016/j.jfa.2009.10.022
Y. Han, G. Lu, and K. Zhao, Discrete Calderón’s identity, atomic decomposition and boundedness criterion of operators on multiparameter Hardy spaces, J. Geom. Anal. 20 (2010), no. 3, 670–689. MR 2610894, DOI 10.1007/s12220-010-9123-6
Jean-Lin Journé, A covering lemma for product spaces, Proc. Amer. Math. Soc. 96 (1986), no. 4, 593–598. MR 826486, DOI 10.1090/S0002-9939-1986-0826486-9
Jean-Lin Journé, Calderón-Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), no. 3, 55–91. MR 836284, DOI 10.4171/RMI/15
Jean-Lin Journé, Two problems of Calderón-Zygmund theory on product-spaces, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 1, 111–132. MR 949001, DOI 10.5802/aif.1125
Michael T. Lacey, Stefanie Petermichl, Jill C. Pipher, and Brett D. Wick, Multiparameter Riesz commutators, Amer. J. Math. 131 (2009), no. 3, 731–769. MR 2530853, DOI 10.1353/ajm.0.0059
Kent G. Merryfield, On the area integral, Carleson measures and $H^p$ in the polydisc, Indiana Univ. Math. J. 34 (1985), no. 3, 663–685. MR 794581, DOI 10.1512/iumj.1985.34.34035
Camil Muscalu, Jill Pipher, Terence Tao, and Christoph Thiele, Bi-parameter paraproducts, Acta Math. 193 (2004), no. 2, 269–296. MR 2134868, DOI 10.1007/BF02392566
Camil Muscalu, Jill Pipher, Terence Tao, and Christoph Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam. 22 (2006), no. 3, 963–976. MR 2320408, DOI 10.4171/RMI/480
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095