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Some estimates for commutators of Littlewood-Paley \(g\)-functions. (English) Zbl 1490.42023

Author’s abstract: The aim of this paper is to establish the boundedness of the commutator \([b,\dot{g}_r]\) generated by Littlewood-Paley \(g\)-functions \(\dot{g}_r\) and \(b\in \mathrm{RBMO}(\mu)\) on non-homogeneous metric measure space. Under assumption that \(\lambda\) satisfies \(\varepsilon\)-weak reverse doubling condition, the author proves that \([b,\dot{g}_r]\) is bounded from Lebesgue spaces \(L^p (\mu)\) into Lebesgue spaces \(L^p (\mu)\) for \(p\in (1,\infty)\) and also bounded from spaces \(L^1 (\mu)\) into spaces \(L^{1,\infty} (\mu)\). Furthermore, the boundedness of \([b,\dot{g}_r]\) on Morrey space \(M_q^p (\mu)\) and on generalized Morrey \(L^{p,\phi} (\mu)\) is obtained.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
30L99 Analysis on metric spaces
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[1] Y. Sawano , Generalized Morrey spaces for non-doubling measures, Nonlinear Differ. Equ. Appl. 15 (2008), no. 4-5, 413-425, . · Zbl 1173.42317 · doi:10.1007/s00030-008-6032-5
[2] Y. Sawano and H. Tanaka , Morrey spaces for non-doubling measures, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1535-1544, . · Zbl 1129.42403 · doi:10.1007/s10114-005-0660-z
[3] X. Tolsa , Littlewood-Paley theory and the T(1) theorem with non-doubling measures, Adv. Math. 164 (2001), no. 1, 57-116, . · Zbl 1015.42010 · doi:10.1006/aima.2001.2011
[4] X. Tolsa , BMO, H1 , and Calderón-Zygmund operators for non-doubling measures, Math. Ann. 319 (2001), no. 1, 89-149, . · Zbl 0974.42014 · doi:10.1007/PL00004432
[5] G. Lu and J. Zhou , Estimates for fractional type Marcinkiewicz integrals with non-doubling measures, J. Inequal. Appl. 2014 (2014), 285, . · Zbl 1323.42015 · doi:10.1186/1029-242X-2014-285
[6] R. R. Coifman and G. Weiss , Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Springer-Verlag, Berlin-New York, 1971. · Zbl 0224.43006
[7] R. R. Coifman and G. Weiss , Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569-645, . · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[8] T. Hytönen , A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa, Publ. Mat. 54 (2010), no. 2, 485-504, . · Zbl 1246.30087 · doi:10.5565/PUBLMAT_54210_10
[9] Y. Cao and J. Zhou , Morrey spaces for nonhomogeneous metric measure spaces, Abstr. Appl. Anal. 2013 (2013), 196459, . · Zbl 1292.42013 · doi:10.1155/2013/196459
[10] X. Fu and J. Zhao , Endpoint estimates of generalized homogeneous Littlewood-Paley g -functions over non-homogeneous metric measure spaces, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 9, 1035-1074, . · Zbl 1351.42022 · doi:10.1007/s10114-016-5059-5
[11] G. Lu and S. Tao , Generalized homogeneous Littlewood-Paley g -function on some function spaces, Bull. Malay. Math. Sci. Soc. 44 (2021), no. 1, 17-34, . · Zbl 1457.42032 · doi:10.1007/s40840-020-00934-7
[12] T. Hytönen , D. Yang , and D. Yang , The Hardy space H1 on non-homogeneous metric measure spaces, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 1, 9-31, . · Zbl 1250.42076 · doi:10.1017/S0305004111000776
[13] C. Tan and J. Li , Littlewood-Paley theory on metric spaces with non-doubling measures and its applications, Sci. China Math. 58 (2015), no. 5, 983-1004, . · Zbl 1320.42012 · doi:10.1007/s11425-014-4950-8
[14] G. Lu and S. Tao , Generalized Morrey spaces over non-homogeneous metric measure spaces, J. Aust. Math. Soc. 103 (2017), no. 2, 268-278, . · Zbl 1376.42024 · doi:10.1017/S1446788716000483
[15] T. A. Bui and X. T. Duong , Hardy spaces, regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces, J. Geom. Anal. 23 (2013), no. 2, 895-932, . · Zbl 1267.42013 · doi:10.1007/s12220-011-9268-y
[16] X. Fu , D. Yang , and W. Yuan , Generalized fractional integral and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math. 18 (2014), no. 2, 509-557, . · Zbl 1357.42016 · doi:10.11650/tjm.18.2014.3651
[17] G.-H. Lu , Commutator of bilinear θ -type Calderón-Zygmund operator on Morrey space over non-homogeneous metric measure spaces, Anal. Math. 46 (2020), no. 1, 97-118, . · Zbl 1449.42016 · doi:10.1007/s10476-020-0020-3
[18] G. Lu and S. Tao , Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces, Open Math. 15 (2017), no. 1, 1283-1299, . · Zbl 1380.42018 · doi:10.1515/math-2017-0110
[19] Y. Zhao , H. Lin , and Y. Meng , Endpoint estimates for multilinear fractional integral operators on metric measure spaces, Ann. Funct. Anal. 10 (2019), no. 3, 337-349, . · Zbl 1515.47078 · doi:10.1215/20088752-2018-0033
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