Weighted estimates for commutators of multilinear Hausdorff operators on variable exponent Morrey-Herz type spaces

Duong, Dao Van ; Dung, Kieu Huu ; Chuong, Nguyen Minh

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 833-865

Abstract (VO)

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.

Détail

DOI : 10.21136/CMJ.2020.0566-18

Classification : 42B30, 42B35, 47B38
Keywords: multilinear Hausdorff operator; Hardy-Cesàro operator; commutator; Lipschitz space; central BMO space; Morrey-Herz space; $A_p$ weight; variable exponent

Bibliographie

[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 | MR | JFM

[2] Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258 (2010), 1628-1655. | DOI | MR | JFM

[3] Bandaliev, R. A.: The boundedness of multidimensional Hardy operators in weighted variable Lebesgue spaces. Lith. Math. J. 50 (2010), 249-259. | DOI | MR | JFM

[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 | MR | JFM

[5] Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31 (1982), 7-16. | DOI | MR | JFM

[6] Chuong, N. M.: Pseudodifferential Operators and Wavelets over Real and $p$-adic Fields. Springer, Cham (2018). | DOI | MR | JFM

[7] Chuong, N. M., Duong, D. V., Dung, K. H.: Multilinear Hausdorff operators on some function spaces with variable exponent. Available at , 36 pages. | arXiv

[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 | MR | JFM

[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 | MR | JFM

[10] Coifman, R. R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212 (1975), 315-331. | DOI | MR | JFM

[11] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 (1976), 611-635. | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[15] Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Am. Math. Soc. 77 (1971), 587-588. | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[18] Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics 250, Springer, New York (2009). | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[22] Indratno, S., Maldonado, D., Silwal, S.: A visual formalism for weights satisfying reverse inequalities. Expo. Math. 33 (2015), 1-29. | DOI | MR | JFM

[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 | MR | JFM

[24] John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961), 415-426. | DOI | MR | JFM

[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 | MR | JFM

[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 | MR | JFM

[27] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165 (1972), 207-226. | DOI | MR | JFM

[28] Stein, E. M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43, Princeton University Press, Princeton (1993). | DOI | MR | JFM

[29] Tang, C., Xue, F., Zhou, Y.: Commutators of weighted Hardy operators on Herz-type spaces. Ann. Polon. Math. 101 (2011), 267-273. | DOI | MR | JFM

[30] Wu, J.: Boundedness for commutators of fractional integrals on Herz-Morrey spaces with variable exponent. Kyoto J. Math. 54 (2014), 483-495. | DOI | MR | JFM

[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 | MR | JFM

[32] Xiao, J.: $L^p$ and BMO bounds of weighted Hardy-Littlewood averages. J. Math. Anal. Appl. 262 (2001), 660-666. | DOI | MR | JFM