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Keywords:
multilinear Hausdorff operator; Hardy-Cesàro operator; commutator; Lipschitz space; central BMO space; Morrey-Herz space; $A_p$ weight; variable exponent
Summary:
We establish the boundedness for the commutators of multilinear Hausdorff operators on the product of some weighted Morrey-Herz type spaces with variable exponent with their symbols belonging to both Lipschitz space and central BMO space. By these, we generalize and strengthen some previously known results.
References:
[1] Almeida, A., Drihem, D.: Maximal, potential and singular type operators on Herz spaces with variable exponents. J. Math. Anal. Appl. 394 (2012), 781-795. DOI 10.1016/j.jmaa.2012.04.043 | MR 2927498 | Zbl 1250.42077
[2] Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258 (2010), 1628-1655. DOI 10.1016/j.jfa.2009.09.012 | MR 2566313 | Zbl 1194.46045
[3] Bandaliev, R. A.: The boundedness of multidimensional Hardy operators in weighted variable Lebesgue spaces. Lith. Math. J. 50 (2010), 249-259. DOI 10.1007/s10986-010-9083-3 | MR 2719561 | Zbl 1219.46026
[4] Brown, G., Móricz, F.: Multivariate Hausdorff operators on the spaces $L^p(\mathbb R^n)$. J. Math. Anal. Appl. 271 (2002), 443-454. DOI 10.1016/S0022-247X(02)00128-2 | MR 1923645 | Zbl 1027.47024
[5] Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31 (1982), 7-16. DOI 10.1512/iumj.1982.31.31002 | MR 0642611 | Zbl 0523.42015
[6] Chuong, N. M.: Pseudodifferential Operators and Wavelets over Real and $p$-adic Fields. Springer, Cham (2018). DOI 10.1007/978-3-319-77473-2 | MR 3839303 | Zbl 06863917
[7] Chuong, N. M., Duong, D. V., Dung, K. H.: Multilinear Hausdorff operators on some function spaces with variable exponent. Available at https://arxiv.org/abs/1709.08185 (2017), 36 pages.
[8] Chuong, N. M., Duong, D. V., Hung, H. D.: Bounds for the weighted Hardy-Cesàro operator and its commutator on weighted Morrey-Herz type spaces. Z. Anal. Anwend. 35 (2016), 489-504. DOI 10.4171/ZAA/1575 | MR 3556758 | Zbl 06655842
[9] Chuong, N. M., Hong, N. T., Hung, H. D.: Multilinear Hardy-Cesàro operator and commutator on the product of Morrey-Herz spaces. Anal. Math. 43 (2017), 547-565. DOI 10.1007/s10476-017-0502-0 | MR 3738360 | Zbl 1399.47125
[10] Coifman, R. R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212 (1975), 315-331. DOI 10.1090/S0002-9947-1975-0380244-8 | MR 0380244 | Zbl 0324.44005
[11] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 (1976), 611-635. DOI 10.2307/1970954 | MR 0412721 | Zbl 0326.32011
[12] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York (2013). DOI 10.1007/978-3-0348-0548-3 | MR 3026953 | Zbl 1268.46002
[13] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017, Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[14] Diening, L., Růžička, M: Calderón-Zygmund operators on generalized Lebesgue spaces $L^{p(\cdot)}$ and problems related to fluid dynamics. J. Reine Angew. Math. 563 (2003), 197-220. DOI 10.1515/crll.2003.081 | MR 2009242 | Zbl 1072.76071
[15] Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Am. Math. Soc. 77 (1971), 587-588. DOI 10.1090/S0002-9904-1971-12763-5 | MR 0280994 | Zbl 0229.46051
[16] Fu, Z. W., Gong, S. L., Lu, S. Z., Yuan, W.: Weighted multilinear Hardy operators and commutators. Forum Math. 27 (2015), 2825-2851. DOI 10.1515/forum-2013-0064 | MR 3393380 | Zbl 1331.42016
[17] Fu, Z. W., Liu, Z. G., Lu, S. Z.: Commutators of weighted Hardy operators on $\mathbb R^n$. Proc. Am. Math. Soc. 137 (2009), 3319-3328. DOI 10.1090/S0002-9939-09-09824-4 | MR 2515401 | Zbl 1174.42018
[18] Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics 250, Springer, New York (2009). DOI 10.1007/978-0-387-09434-2 | MR 2463316 | Zbl 1158.42001
[19] Guliyev, V. S., Hasanov, J. J., Samko, S. G.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107 (2010), 285-304. DOI 10.7146/math.scand.a-15156 | MR 2735097 | Zbl 1213.42077
[20] Hung, H. D., Ky, L. D.: New weighted multilinear operators and commutators of Hardy-Cesàro type. Acta Math. Sci., Ser. B, Engl. Ed. 35 (2015), 1411-1425. DOI 10.1016/S0252-9602(15)30063-1 | MR 3413504 | Zbl 1349.42050
[21] Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for $A_{\infty}$ weights on spaces of homogeneous type. J. Funct. Anal. 263 (2012), 3883-3899. DOI 10.1016/j.jfa.2012.09.013 | MR 2990061 | Zbl 1266.42045
[22] Indratno, S., Maldonado, D., Silwal, S.: A visual formalism for weights satisfying reverse inequalities. Expo. Math. 33 (2015), 1-29. DOI 10.1016/j.exmath.2013.12.008 | MR 3310925 | Zbl 1344.42021
[23] Izuki, M.: Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend. Circ. Mat. Palermo (2) 59 (2010), 461-472. DOI 10.1007/s12215-010-0034-y | MR 2745523 | Zbl 1206.42011
[24] John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961), 415-426. DOI 10.1002/cpa.3160140317 | MR 0131498 | Zbl 0102.04302
[25] Karlovich, A. Y., Lerner, A. K.: Commutators of singular integrals on generalized $L^p$ spaces with variable exponent. Publ. Mat., Barc. 49 (2005), 111-125. DOI 10.5565/PUBLMAT_49105_05 | MR 2140202 | Zbl 1129.42359
[26] Lu, Y., Zhu, Y. P.: Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents. Acta Math. Sin., Engl. Ser. 30 (2014), 1180-1194. DOI 10.1007/s10114-014-3410-2 | MR 3226766 | Zbl 1304.42042
[27] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165 (1972), 207-226. DOI 10.1090/S0002-9947-1972-0293384-6 | MR 0293384 | Zbl 0236.26016
[28] Stein, E. M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43, Princeton University Press, Princeton (1993). DOI 10.1515/9781400883929 | MR 1232192 | Zbl 0821.42001
[29] Tang, C., Xue, F., Zhou, Y.: Commutators of weighted Hardy operators on Herz-type spaces. Ann. Polon. Math. 101 (2011), 267-273. DOI 10.4064/ap101-3-6 | MR 2799588 | Zbl 1230.42023
[30] Wu, J.: Boundedness for commutators of fractional integrals on Herz-Morrey spaces with variable exponent. Kyoto J. Math. 54 (2014), 483-495. DOI 10.1215/21562261-2693397 | MR 3263547 | Zbl 1310.42009
[31] Wu, J.-L., Zhao, W.-J.: Boundedness for fractional Hardy-type operator on variable-exponent Herz-Morrey spaces. Kyoto J. Math. 56 (2016), 831-845. DOI 10.1215/21562261-3664932 | MR 3568643 | Zbl 1354.42025
[32] Xiao, J.: $L^p$ and BMO bounds of weighted Hardy-Littlewood averages. J. Math. Anal. Appl. 262 (2001), 660-666. DOI 10.1006/jmaa.2001.7594 | MR 1859331 | Zbl 1009.42013
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