Boundedness of Vector Valued Bilinear Calderón-Zygmund Operators on Products of Weighted Herz-Morrey Spaces with Variable Exponents

Abstract

In this paper, the authors obtain the boundedness of vector valued bilinear Calderón-Zygmund operators on products of weighted Herz-Morrey spaces with variable exponents.

Keywords

Bilinear Calderón-Zygmund operator / Vector valued inequality / Muckenhoupt weight / Variable exponent / Herz-Morrey space

References

[1] Almeida A, Hästö P. Besov spaces with variable smoothness and integrability. J. Funct. Anal., 2010, 258: 1628-1655

[2] Bui T A, Duong X T. Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull. Sci. Math., 2013, 137(1): 63-75

[3] Coifman R R, Meyer Y. On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc, 1975, 212: 315-331

[4] Cruz-Uribe D, Diening L, Hästö P. The maximal operator on weighted variable Lebesgue spaces. Fract. Calc. Appl. Anal., 2011, 14(3): 361-374

[5] Cruz-Uribe D, Fiorenza A, Neugebauer C J. Weighted norm inequalities for the maximal operator on variable Lebesgue spaces. J. Math. Anal. Appl., 2012, 394(2): 744-760

[6] Cruz-Uribe D, Martell J M, Pérez C. Extrapolation from A _∞ weights and applications. J. Funct. Anal., 2004, 213(2): 412-439

[7] Cruz-Uribe D, Martell J M, Pérez C. Weights, Extrapolation and the Theory of Rubio de Francia, 2011, Basel: Birkhäuser

[8] Cruz-Uribe D, Wang L A D. Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. Trans. Amer. Math. Soc, 2016, 369(2): 1205-1235

[9] Grafakos L, Kaiton N. Multilinear Calderón-Zygmund operators on Hardy spaces. Collect. Math., 2001, 52: 169-179

[10] Grafakos L, Torres R H. Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J., 2002, 51: 1261-1276

[11] Grafakos L, Torres R H. Multilinear Calderón-Zygmund theory. Adv. Math., 2002, 165(1): 124-164

[12] Grafakos, L. and Torres, R. H., On multilinear singular integrals of Calderón-Zygmund type, Publ. Mat., Extra, 2002, 57–91.

[13] Hu G E, Meng Y. Multilinear Calderón-Zygmund operator on products of Hardy spaces. Acta Math. Sinica (Engl. Ser), 2012, 28(2): 281-294

[14] Hu Y Z, Xu J S. Multilinear Calderón-Zygmund operators and their commutators with BMO functions in Herz-Morrey spaces with variable smoothness and integrability. Commun. Math. Res, 2017, 33(3): 238-258

[15] Huang A W, Xu J S. Multilinear singular integral and commutators in variable exponent Lebesgue space. Appl. Math. J. Chinese Univ. Ser. B, 2010, 25(1): 69-77

[16] Izuki M. Remarks on Muckenhoupt weights with variable exponent. J. Anal. Appl., 2013, 11(1): 27-41

[17] Izuki M, Nakai E, Sawano Y. Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent. Ann. Acad. Sci. Fenn. Math., 2015, 40(2): 551-571

[18] Izuki M, Noi T. An intrinsic square function on weighted Herz spaces with variable exponent. J. Math. Inequal., 2016, 11(3): 799-816

[19] Izuki, M. and Noi, T., Boundedness of fractional integrals on weighted Herz spaces with variable exponent, J. Inequal. Appl., 2016, 2016, Article ID 199.

[20] Izuki M, Noi T. Two weighted Herz spaces with variable exponents. Bull. Malays. Math. Sci. Soc., 2020, 43: 169-200

[21] Lu Y, Zhu Y P. Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents. Acta Math. Sinica, 2014, 30(7): 1180-1194

[22] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function. Tran. Amer. Math. Soc., 1972, 165: 207-226

[23] Pérez C, Torres R H. Sharp maximal function estimates for multilinear singular integrals. Publ. Mat., 2002, 46(2): 229-274

[24] Pérez C, Torres R H. Sharp maximal function estimates for multilinear singular integrals. Contemp. Math., 2003, 320: 323-331

[25] Pérez C, Trujillo-González R. Sharp weighted estimates for vector-valued singular integral operators and commutators. Tohoku Math. J., 2003, 55(1): 109-129

[26] Sawano Y. Theory of Besov Spaces, 2018, Singapore: Springer-Verlag

[27] Shen C H, Xu J S. A vector-valued estimate of multilinear Calderón-Zygmund operators in Herz-Morrey spaces with variable exponents. Hokkaido Math. J., 2017, 46(3): 351-380

[28] Wang, P. and Liu, Z., Weighted norm inequalities for multilinear Calderón-Zygmund operators in generalized Morrey spaces, J. Inequal. Appl., 2017, 2017, Article ID 48.

[29] Wang, S. R. and Xu, J. S., Weighted norm inequality for bilinear CalderónCZygmund operators on HerzC-Morrey spaces with variable exponents, J. Inequal. Appl., 2019, 2019, Article ID 251.

[30] Wang W, Xu J S. Multilinear Calderón-Zygmund operators and their commutators with BMO functions in variable exponent Morrey spaces. Front. Math. China, 2017, 12(5): 1235-1246

[31] Xu J S. Boundedness of multilinear singular integrals for non-doubling measures. J. Math. Anal. Appl., 2007, 327(1): 471-480

[32] Xu J S. Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures. Sci. China Ser. A, 2007, 50(3): 361-376