On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces

  • M. Qu School Math. and Statistics, Anhui Normal Univ., Wuhu, China
  • L. Wang School Math. and Physics, Anhui Polytech. Univ., Wuhu, China


UDC 517.5

In the framework of variable exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The approach is based on the theory of variable exponent and on generalization of the BMO-norms.


A. Almeida, D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J.Math. Anal. and Appl., 394, 781 – 795 (2012) https://doi.org/10.1016/j.jmaa.2012.04.043

Y. Chen, Y. Ding, $L^p$ boundedness of the commutators of Marcinkiewicz integrals with rough kernels, Forum Math., 27, 2087 – 2111 (2015) https://doi.org/10.1515/forum-2013-0041

Y. Chen, S. Levine, R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, 1383 – 1406 (2006) https://doi.org/10.1137/050624522

D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue spaces: foundations and harmonic analysis, App. and Numer. Harmon. Anal., Birkhauser, Basel 2013. x+312 pp. ISBN: 978-3-0348-0547-6; 978-3-0348-0548-3 (2013) https://doi.org/10.1007/978-3-0348-0548-3

D. Cruz-Uribe, SFO, A. Fiorenza, J. Martell, C. Perez, ´ The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math., 31, 239 – 264 (2006).

L. Diening, P. Harjulehto, P. Hästö, M. Růžička, ˇ Lebesgue and sobolev spaces with variable exponents, Lect. Notes Math., 2017 x+509 pp. ISBN: 978-3-642-18362-1 (2011)

Y. Ding, D. Fan, Y. Pan, $L^p$-boundedness of Marcinkiewicz integrals with Hardy space function kernels, Acta Math. Sin. (Engl. Ser.), 16, 593 – 600 (2000) https://doi.org/10.1007/s101140000015

Y. Ding, S. Lu, K. Yabuta, On commutators of Marcinkiewicz integrals with rough kernel, J. Math. Anal. and Appl., 275, 60 – 68 (2002) https://doi.org/10.1016/S0022-247X(02)00230-5

K.-P. Ho, The fractional integral operators on Morrey spaces with variable exponent on unbounded domains, Math. Inequal. and Appl., 16, 363 – 373 (2013) https://doi.org/10.7153/mia-16-27

K.-P. Ho, Vector-valued singular integral operators on Morrey type spaces and variable Triebel – Lizorkin – Morrey spaces, Ann. Acad. Sci. Fenn. Math., 37, 375 – 406 (2012) https://doi.org/10.5186/aasfm.2012.3746

P. Harjulehto, P. Hästö, Ú. V. Lê, M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72, 4551 – 4574 (2010) https://doi.org/10.1016/j.na.2010.02.033

M. Izuki, Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent, Rend. Circ. Mat. Palermo, 59, 461 – 472 (2010) https://doi.org/10.1007/s12215-010-0034-y

O. Kova´cik, J. Rákosník, On spaces $ L^{p (x)} $ and $ W^{k, p (x)} $, Czechoslovak Math. J., 41, 592 – 618 (1991) https://dml.cz/handle/10338.dmlcz/102493

F. Liu, Integral operators of Marcinkiewicz type on Triebel-Lizorkin spaces, Math. Nachr. 290, 75 – 96 (2017) https://doi.org/10.1002/mana.201500374

F. Liu, On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces, Math. Inequal. and Appl., 20, 515 – 535 (2017) https://doi.org/10.7153/mia-20-35

F. Liu, Z. Fu, Y. Zheng, Q. Yuan, A rough Marcinkiewicz integral along smooth curves, J. Nonlinear Sci. and Appl., 9, 4450 – 4464 (2016) https://doi.org/10.22436/jnsa.009.06.84

F. Liu, H. Wu, $L^p$ bounds for Marcinkiewicz integrals associated to homogeneous mappings, Monatsh. Math., 181, 875 – 906 (2016) https://doi.org/10.1007/s00605-016-0968-z

F. Liu, H. Wu, D. Zhang, $L^p$ bounds for parametric Marcinkiewicz integrals with mixed homogeneity, Math. Inequal. and Appl., 18, 453 – 469 (2015) https://doi.org/10.7153/mia-18-34

S. Lu, L. Xu, Boundedness of rough singular intergral operators on the homogeneous Morrey – Herz spaces, Hokkaido Math. J., 34, 299 – 314 (2005) https://doi.org/10.14492/hokmj/1285766224

Y. Lu, Y. Zhu, Boundedness of some sublinear operators and commutators on Morrey – Herz spaces with variable exponents, Czechoslovak Math. J., 64(139), no. 4, 969 – 987 (2014) https://doi.org/10.1007/s10587-014-0147-0

E. Nakai, Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal., 262, 3665 – 3748 (2012) https://doi.org/10.1016/j.jfa.2012.01.004

H. Rafeiro, S. Samko, On maximal and potential operators with rough kernels in variable exponent spaces, Rend. Lincei Mat. Appl., 27, 309 – 325 (2016) https://doi.org/10.4171/RLM/736

E. M. Stein, On the functions of Littlewood – Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88, 430 – 466 (1958) https://doi.org/10.2307/1993226

A. Torchinsky, Real-variable methods in harmonic analysis, Pure and Appl. Math., 123, Academic Press, Inc., Orlando, FL, 1986 (2012).

A. Torchinsky, S. Wang, A note on the Marcinkiewicz integral, Colloq. Math., 60/61, 235 – 243 xii+462 pp. ISBN: 0-12-695460-7; 0-12-695461-5 (1990).

S. Tao, L. Li, Boundedness of Marcinkiewicz integrals and commutators on Morrey spaces with variable exponents (Chinese) ; translated from Chinese Ann. Math. Ser. A 37 (2016), no. 1, 55--70 Chinese J. Contemp. Math. 37 (2016), no. 1, 53--68.

J. Tan, J. Zhao, Fractional integrals on variable Hardy – Morrey spaces, Acta Math. Hungar., 148, 174 – 190 (2016) https://doi.org/10.1007/s10474-015-0571-0

H. Wang, Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent, Czechoslovak Math. J.,66(141), 251 – 269 (2016) https://doi.org/10.1007/s10587-016-0254-1

L. Wang, M. Qu, L. Shu, Boundedness of rough singular integral operators and commutators on Morrey – Herz spaces with variable exponents, J. Inequal. and Appl., 2016 (2016) https://doi.org/10.1186/s13660-016-1161-6

D. Yang, C. Zhuo, W. Yuan, Triebel-Lizorkin type spaces with variable exponents, Banach J. Math. Anal., 9, 146 – 202 (2015) https://doi.org/10.15352/bjma/09-4-9

D. Yang, C. Zhuo, W. Yuan, Besov-type spaces with variable smoothness and integrability, J. Funct. Anal., 269, 1840 – 1898 (2015) https://doi.org/10.1016/j.jfa.2015.05.016

P. Zhang, J. Wu, Commutators for the maximal function on Lebesgue spaces with variable exponent, Math. Inequal. and Appl., 17, 1375 – 1386 (2014) https://doi.org/10.7153/mia-17-101

How to Cite
QuM., and WangL. “On the Commutator of Marcinkiewicz Integrals With Rough Kernels in Variable Morrey Type Spaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 7, July 2020, pp. 928-44, doi:10.37863/umzh.v72i7.6023.
Research articles