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http://dx.doi.org/10.4134/BKMS.b201019

THE CHARACTERISATION OF BMO VIA COMMUTATORS IN VARIABLE LEBESGUE SPACES ON STRATIFIED GROUPS  

Liu, Dongli (Department of Mathematics and Physics Shijiazhuang Tiedao University, School of Mathematical Sciences Beijing Normal University)
Tan, Jian (School of Science Nanjing University of Posts and Telecommunications)
Zhao, Jiman (School of Mathematical Sciences Beijing Normal University)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.3, 2022 , pp. 547-566 More about this Journal
Abstract
Let T be a bilinear Calderón-Zygmund operator, $b{\in}U_q>_1L^q_{loc}(G)$. We firstly obtain a constructive proof of the weak factorisation of Hardy spaces. Then we establish the characterization of BMO spaces by the boundedness of the commutator [b, T]j in variable Lebesgue spaces.
Keywords
Bilinear Calderon-Zygmund operators; variable Lebesgue spaces; stratified groups;
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1 E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665-3748. https://doi.org/10.1016/j.jfa.2012.01.004   DOI
2 W. Orlicz, Uber konjugierte Exponentenfolgen, Stud. Math. 3 (1931), 200-211.   DOI
3 C. Perez and R. H. Torres, Sharp maximal function estimates for multilinear singular integrals, in Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), 323-331, Contemp. Math., 320, Amer. Math. Soc., Providence, RI, 2003. https://doi.org/10.1090/conm/320/05615   DOI
4 J. Tan, Z. Liu, and J. Zhao, On some multilinear commutators in variable Lebesgue spaces, J. Math. Inequal. 11 (2017), no. 3, 715-734. https://doi.org/10.7153/jmi2017-11-57   DOI
5 J. Tan and J. Zhao, Rough fractional integrals and its commutators on variable Morrey spaces, C. R. Math. Acad. Sci. Paris 353 (2015), no. 12, 1117-1122. https://doi.org/10.1016/j.crma.2015.09.024   DOI
6 A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. (2) 30 (1978), no. 1, 163-171. https://doi.org/10.2748/tmj/1178230105   DOI
7 H. Wang and Z. Liu, The wavelet characterization of Herz-type Hardy spaces with variable exponent, Ann. Funct. Anal. 3 (2012), no. 1, 128-141. https://doi.org/10.15352/afa/1399900030   DOI
8 Y. Zhu and L. Liu, Weighted sharp boundedness for multilinear commutators of singular integral on spaces of homogeneous type, Vietnam J. Math. 39 (2011), no. 4, 381-390.
9 D. Yang, C. Zhuo, and W. Yuan, Triebel-Lizorkin type spaces with variable exponents, Banach J. Math. Anal. 9 (2015), no. 4, 146-202. https://doi.org/10.15352/bjma/09-4-9   DOI
10 X. Fu, D. Yang, and W. Yuan, Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math. 18 (2014), no. 2, 509-557. https://doi.org/10.11650/tjm.18.2014.3651   DOI
11 L. Liu, D. Chang, X. Fu, and D. Yang, Endpoint boundedness of commutators on spaces of homogeneous type, Appl. Anal. 96 (2017), no. 14, 2408-2433. https://doi.org/10.1080/00036811.2017.1341628   DOI
12 D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-0348-0548-3   DOI
13 R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611-635. https://doi.org/10.2307/1970954   DOI
14 D. Cruz-Uribe and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447-493. https://doi.org/10.1512/iumj.2014.63.5232   DOI
15 G. Lu, S. Lu, and D. Yang, Singular integrals and commutators on homogeneous groups, Anal. Math. 28 (2002), no. 2, 103-134. https://doi.org/10.1023/A:1016568918973   DOI
16 M. Bramanti, Commutators of integral operators with positive kernels, Matematiche (Catania) 49 (1994), no. 1, 149-168.
17 M. Bramanti and M. C. Cerutti, Commutators of singular integrals and fractional integrals on homogeneous spaces, in Harmonic analysis and operator theory (Caracas, 1994), 81-94, Contemp. Math., 189, Amer. Math. Soc., Providence, RI, 1995. https://doi.org/10.1090/conm/189/02257   DOI
18 L. Chaffee, Commutators of multilinear singular integral operators with pointwise multiplication, Thesis (Ph.D.), University of Kansas, 2015.
19 S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), no. 1, 7-16. https://doi.org/10.1512/iumj.1982.31.31002   DOI
20 X. Chen and Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), no. 2, 355-373. https://doi.org/10.1016/j.jmaa.2009.08.022   DOI
21 D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Perez, The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239-264.
22 X. T. Duong, H. Li, J. Li, and B. D. Wick, Lower bound of Riesz transform kernels and commutator theorems on stratified nilpotent Lie groups, J. Math. Pures Appl. (9) 124 (2019), 273-299. https://doi.org/10.1016/j.matpur.2018.06.012   DOI
23 J. Fang and J. Zhao, Variable Hardy spaces on the Heisenberg group, Anal. Theory Appl. 32 (2016), no. 3, 242-271. https://doi.org/10.4208/ata.2016.v32.n3.4   DOI
24 G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, 28, Princeton University Press, Princeton, NJ, 1982.
25 L. Grafakos and R. H. Torres, Multilinear Calderon-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124-164. https://doi.org/10.1006/aima.2001.2028   DOI
26 S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), no. 2, 263-270. https://doi.org/10.1007/BF02386000   DOI
27 Z. Guo, P. Li, and L. Peng, Lp boundedness of commutators of Riesz transforms associated to Schrodinger operator, J. Math. Anal. Appl. 341 (2008), no. 1, 421-432. https://doi.org/10.1016/j.jmaa.2007.05.024   DOI
28 P. A. Hasto, Local-to-global results in variable exponent spaces, Math. Res. Lett. 16 (2009), no. 2, 263-278. https://doi.org/10.4310/MRL.2009.v16.n2.a5   DOI
29 G. Hu, Y. Meng, and D. Yang, Multilinear commutators of singular integrals with non doubling measures, Integral Equations Operator Theory 51 (2005), no. 2, 235-255. https://doi.org/10.1007/s00020-003-1251-y   DOI
30 P. Li and L. Peng, Lp boundedness of commutator operator associated with Schrodinger operators on Heisenberg group, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 2, 568-578. https://doi.org/10.1016/S0252-9602(12)60039-3   DOI
31 J. Li and B. D. Wick, Weak factorizations of the Hardy space H1(ℝn) in terms of multilinear Riesz transforms, Canad. Math. Bull. 60 (2017), no. 3, 571-585. https://doi.org/10.4153/CMB-2017-033-9   DOI
32 Y. Liang, L. D. Ky, and D. Yang, Weighted endpoint estimates for commutators of Calderon-Zygmund operators, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5171-5181. https://doi.org/10.1090/proc/13130   DOI
33 H. Lin, S. Wu, and D. Yang, Boundedness of certain commutators over non-homogeneous metric measure spaces, Anal. Math. Phys. 7 (2017), no. 2, 187-218. https://doi.org/10.1007/s13324-016-0136-6   DOI
34 Z. Liu and S. Lu, Two-weight weak-type norm inequalities for the commutators of fractional integrals, Integral Equations Operator Theory 48 (2004), no. 3, 397-409. https://doi.org/10.1007/s00020-002-1237-1   DOI
35 O. Kovacik and J. Rakosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(116) (1991), no. 4, 592-618.
36 D. Liu, J. Tan, and J. Zhao, Multilinear commutators in variable Lebesgue spaces on stratified groups, in Analysis of pseudo-differential operators, 97-120, Trends Math, Birkhauser/Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-05168-6_5   DOI
37 H. Liu and L. Tang, Compactness for higher order commutators of oscillatory singular integral operators, Internat. J. Math. 20 (2009), no. 9, 1137-1146. https://doi.org/10.1142/S0129167X09005698   DOI