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Czechoslovak Mathematical Journal, Vol. 72, No. 3, pp. 709-734, 2022

Weighted multi-parameter mixed Hardy spaces and their applications

Wei Ding, Yun Xu, Yueping Zhu

Received March 27, 2021.   Published online January 26, 2022.
Abstract:  Applying discrete Calderón's identity, we study weighted multi-parameter mixed Hardy space $H^p_{\rm mix}(\omega,\mathbb{R}^{n_1} \times \mathbb{R}^{n_2})$. Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on $H^p_{\rm mix}(\omega,\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ of operators in mixed Journé's class.
Keywords:  weight; multi-parameter; mixed Hardy spaces; singular integral operator
Classification MSC:  42B35, 42B30, 42B25, 42B20
DOI:  10.21136/CMJ.2022.0115-21
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References:
[1] R. R. Coifman: A real variable characterization of $H^p$. Stud. Math. 51 (1974), 269-274. DOI 10.4064/sm-51-3-269-274 | MR 0358318 | Zbl 0289.46037
[2] R. R. Coifman, G. Weiss: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83 (1977), 569-645. DOI 10.1090/S0002-9904-1977-14325-5 | MR 0447954 | Zbl 0358.30023
[3] D. Cruz-Uribe, J. M. Martell, C. Pérez: Sharp weighted estimates for classical operators. Adv. Math. 229 (2012), 408-441. DOI 10.1016/j.aim.2011.08.013 | MR 2854179 | Zbl 1236.42010
[4] Y. Ding, Y. Han, G. Lu, X. Wu: Boundedness of singular integrals on multiparameter weighted Hardy spaces $H^p_w({\mathbb R}^n\times {\mathbb R}^m)$. Potential Anal. 37 (2012), 31-56. DOI 10.1007/s11118-011-9244-y | MR 2928237 | Zbl 1271.42017
[5] W. Ding, G. Lu, Y. Zhu: Multi-parameter local Hardy spaces. Nonlinear Anal., Theory Methods Appl., Ser. A 184 (2019), 352-380. DOI 10.1016/j.na.2019.02.014 | MR 3925053 | Zbl 1418.42038
[6] W. Ding, M. Qin, Y. Zhu: The boundedness on mixed Hardy spaces. J. Funct. Spaces 2020 (2020), Article ID 5341674, 12 pages. DOI 10.1155/2020/5341674 | MR 4078744 | Zbl 1436.42018
[7] W. Ding, Y. Zhu: Mixed Hardy spaces and their applications. Acta Math. Sci., Ser. B, Engl. Ed. 40 (2020), 945-969. DOI 10.1007/s10473-020-0405-1 | MR 4108498
[8] C. L. Fefferman, E. M. Stein: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. DOI 10.1007/BF02392215 | MR 0447953 | Zbl 0257.46078
[9] R. Fefferman: Strong differentiation with respect to measures. Am. J. Math. 103 (1981), 33-40. DOI 10.2307/2374188 | MR 0601461 | Zbl 0475.42019
[10] R. Fefferman: The atomic decomposition of $H^1$ in product spaces. Adv. Math. 55 (1985), 90-100. DOI 10.1016/0001-8708(85)90006-4 | MR 0772072 | Zbl 0606.42016
[11] R. Fefferman: Calderón-Zygmund theory for product domains: $H^p$ spaces. Proc. Natl. Acad. Sci. USA 83 (1986), 840-843. DOI 10.1073/pnas.83.4.840 | MR 0828217 | Zbl 0602.42023
[12] R. Fefferman: Harmonic analysis on product spaces. Ann. Math. (2) 126 (1987), 109-130. DOI 10.2307/1971346 | MR 0898053 | Zbl 0644.42017
[13] R. Fefferman, E. M. Stein: Singular integrals on product spaces. Adv. Math. 45 (1982), 117-143. DOI 10.1016/S0001-8708(82)80001-7 | MR 0664621 | Zbl 0517.42024
[14] G. B. Folland, E. M. Stein: Hardy Spaces on Homogeneous Groups. Mathematical Notes 28. Princeton University Press, Princeton (1982). DOI 10.2307/j.ctv17db3q0 | MR 0657581 | Zbl 0508.42025
[15] J. Garcia-Cuerva: Weighted $H^p$ spaces. Diss. Math. 162 (1979), 63 pages. MR 0549091 | Zbl 0434.42023
[16] J. Garcia-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116. North-Holland, Amsterdam (1985). DOI 10.1016/s0304-0208(08)x7154-3 | MR 0807149 | Zbl 0578.46046
[17] D. Goldberg: A local version of real Hardy spaces. Duke Math. J. 46 (1979), 27-42. DOI 10.1215/S0012-7094-79-04603-9 | MR 0523600 | Zbl 0409.46060
[18] L. Grafakos: Classical and Modern Fourier Analysis. Pearson/Prentice Hall, Upper Saddle River (2004). MR 2449250 | Zbl 1148.42001
[19] R. F. Gundy, E. M. Stein: $H^p$ theory for the poly-disk. Proc. Natl. Acad. Sci. USA 76 (1979), 1026-1029. DOI 10.1073/pnas.76.3.1026 | MR 0524328 | Zbl 0405.32002
[20] R. F. Gundy, R. L. Wheeden: Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series. Stud. Math. 49 (1974), 107-124. DOI 10.4064/sm-49-2-107-124 | MR 0352854 | Zbl 0271.28002
[21] Y. Han, M.-Y. Lee, C.-C. Lin, Y.-C. Lin: Calderón-Zygmund operators on product Hardy spaces. J. Funct. Anal. 258 (2010), 2834-2861. DOI 10.1016/j.jfa.2009.10.022 | MR 2593346 | Zbl 1197.42006
[22] Y. Han, C. Lin, G. Lu, Z. Ruan, E. T. Sawyer: Hardy spaces associated with different homogeneities and boundedness of composition operators. Rev. Mat. Iberoam. 29 (2013), 1127-1157. DOI 10.4171/RMI/751 | MR 3148598 | Zbl 1291.42018
[23] Y. Han, G. Lu, Z. Ruan: Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces. J. Funct. Anal. 264 (2013), 1238-1268. DOI 10.1016/j.jfa.2012.12.006 | MR 3010020 | Zbl 1268.42024
[24] Y. Han, G. Lu, Z. Ruan: Boundedness of singular integrals in Journé's class on weighted multiparameter Hardy spaces. J. Geom. Anal. 24 (2014), 2186-2228. DOI 10.1007/s12220-013-9421-x | MR 3261735 | Zbl 1302.42024
[25] Y. Han, G. Lu, K. Zhao: Discrete Calderón's identity, atomic decomposition and boundedness criterion of operators on multiparameter Hardy spaces. J. Geom. Anal. 20 (2010), 670-689. DOI 10.1007/s12220-010-9123-6 | MR 2610894 | Zbl 1193.42090
[26] G. H. Hardy: The mean value of the modulus of an analytic function. Proc. Lond. Math. Soc. (2) 14 (1915), 269-277. DOI 10.1112/plms/s2_14.1.269 | JFM 45.1331.03
[27] J.-L. Journé: Calderón-Zygmund operators on product spaces. Rev. Mat. Iberoam. 1 (1985), 55-91. DOI 10.4171/RMI/15 | MR 0836284 | Zbl 0634.42015
[28] J.-L. Journé: Two problems of Calderón-Zygmund theory on product-spaces. Ann. Inst. Fourier 38 (1988), 111-132. DOI 10.5802/aif.1125 | MR 0949001 | Zbl 0638.47026
[29] R. H. Latter: A characterization of $H^p(\mathbb{R}^n)$ in terms of atoms. Stud. Math. 62 (1978), 93-101. DOI 10.4064/sm-62-1-93-101 | MR 0482111 | Zbl 0398.42017
[30] G. Lu, Y. Zhu: Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters. Acta Math. Sin., Engl. Ser. 29 (2013), 39-52. DOI 10.1007/s10114-012-1402-7 | MR 3001008 | Zbl 1261.42030
[31] Z. Ruan: Weighted Hardy spaces in three-parameter case. J. Math. Anal. Appl. 367 (2010), 625-639. DOI 10.1016/j.jmaa.2010.02.010 | MR 2607286 | Zbl 1198.42015
[32] E. M. Stein, G. Weiss: On the theory of harmonic functions of several variables. I. The theory of $H^p$-spaces. Acta Math. 103 (1960), 25-62. DOI 10.1007/BF02546524 | MR 0121579 | Zbl 0097.28501
[33] D. Yang, Y. Liang, L. D. Ky: Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics 2182. Springer, Cham (2017). DOI 10.1007/978-3-319-54361-1 | MR 3586020 | Zbl 1375.42038
[34] D. Yang, S. Yang: Local Hardy spaces of Musielak-Orlicz type and their applications. Sci. China, Math. 55 (2012), 1677-1720. DOI 10.1007/s11425-012-4377-z | MR 2955251 | Zbl 1266.42055
Affiliations:   Wei Ding (corresponding author), Yun Xu, School of Sciences, Nantong University, Nantong 226007, P. R. China, e-mail: dingwei@ntu.edu.cn, 1134721412@qq.com; Yueping Zhu, Department of Mathematics, Nantong Normal College, Nantong 226010, P. R. China, e-mail: zhuyueping@ntu.edu.cn
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