Boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and Morrey-Herz spaces with variable exponents

Yueping Zhu; Yan Tang; Lixin Jiang; Yueping Zhu; Yan Tang; Lixin Jiang

doi:10.3934/math.2021652

AIMS Mathematics

2021, Volume 6, Issue 10: 11246-11262. doi: 10.3934/math.2021652

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Research article

Boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and Morrey-Herz spaces with variable exponents

1.
Department of Mathematics, Nantong Normal College, Nantong 226010, China
2.
School of Science, Nantong University, Nantong 226010, China

Received: 02 June 2021 Accepted: 28 July 2021 Published: 04 August 2021
MSC : 42B25, 42B35

In this paper, we introduce weighted Morrey-Herz spaces $ M\dot K^{\alpha, \lambda}_{q, p(\cdot)}(w~^{p(\cdot)}) $ with variable exponent $ p(\cdot) $. Then we prove the boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and weighted Morrey-Herz spaces with variable exponents.
- weighted Lebesgue spaces,
- Morrey-Herz spaces,
- variable exponents,
- multilinear Calderón-Zygmund singular operators,
- boundedness
Citation: Yueping Zhu, Yan Tang, Lixin Jiang. Boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and Morrey-Herz spaces with variable exponents[J]. AIMS Mathematics, 2021, 6(10): 11246-11262. doi: 10.3934/math.2021652

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Abstract

In this paper, we introduce weighted Morrey-Herz spaces $ M\dot K^{\alpha, \lambda}_{q, p(\cdot)}(w~^{p(\cdot)}) $ with variable exponent $ p(\cdot) $. Then we prove the boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and weighted Morrey-Herz spaces with variable exponents.

References

[1]	L. Grafakos, R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math., 165 (2002), 124-164. doi: 10.1006/aima.2001.2028
[2]	G. P. Curbera, J. García-Cuerva, J. M. Martell, C. Pérez, Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals, Adv. Math., 203 (2006), 256-318. doi: 10.1016/j.aim.2005.04.009
[3]	C. Pérez, R. H. Torres, Sharp maximal function estimates for multinear singular integrals, Contemp. Math., 320 (2003), 1-9. doi: 10.1090/conm/320/05594
[4]	X. X. Tao, H. H. Zhang, On the boundedness of multilinear operators on weighted Herz-Morrey spaces, Taiwan. J. Math., 15 (2011), 1527-1543.
[5]	O. Kováčik, J. Rákosník, On spaces $L~~^{p(x)}$ and $W^{k, p(x)}$, Czech. Math. J., 41 (1991), 592-618. doi: 10.21136/CMJ.1991.102493
[6]	D. Cruz-Uribe, L. A. D. Wang, Extrapolation and weighted norm inequalities in the variable Lebesgue spaces, Trans. Am. Math. Soc., 369 (2017), 1205-1235.
[7]	M. Izuki, T. Noi, Boundedness of fractional integral on weighted Herz spaces with variable exponent, J. Inequal. Appl., 2016 (2016), 199. doi: 10.1186/s13660-016-1142-9
[8]	D. C. Yang, C. Q. Zhuo, W. Yuan, Besov-type spaces with variable smoothness and integrability, J. Funct. Anal., 269 (2015), 1840-1898. doi: 10.1016/j.jfa.2015.05.016
[9]	D. C. Yang, C. Q. Zhuo, W. Yuan, Triebel-Lizorkin type spaces with variable exponents, Banach J. Math. Anal., 9 (2015), 146-202. doi: 10.15352/bjma/09-4-9
[10]	X. J. Yan, D. C. Yang, W. Yuan, C. Q. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal., 271 (2016), 2822-2887. doi: 10.1016/j.jfa.2016.07.006
[11]	L. W. Wang, L. S. Shu, Higher order commutators of fractional integrals on Morrey type spaces with variable exponents, Math. Nachr., 291 (2018), 1437-1449. doi: 10.1002/mana.201600438
[12]	A. W. Huang, J. S. Xu, Multilinear singular integrals and commutators in variable exponent Lebesgue spaces, Appl. Math. J. Chin. Univ., 25 (2010), 69-77. doi: 10.1007/s11766-010-2167-3
[13]	X. X. Tao, X. Yu, H. H. Zhang, Multilinear Calderón Zygmund operators on variable exponent Morrey spaces over domains, Appl. Math. J. Chin. Univ., 26 (2011), 187-197. doi: 10.1007/s11766-011-2704-8
[14]	Z. W. Fu, S. Z. Lu, H. B. Wang, L. G. Wang, Singular integral operators with rough kernels on central Morrey spaces with variable exponent, Ann. Acad. Sci. Fenn. Math., 44 (2019), 505-522. doi: 10.5186/aasfm.2019.4431
[15]	Y. Lu, Y. P. Zhu, Boundedness of multilinear Calderón-Zygmund singular operators on Morrey-Herz spaces with variable exponents, Acta Math. Sin.-English Ser., 30 (2014), 1180-1194. doi: 10.1007/s10114-014-3410-2
[16]	D. Cruz-Uribe, L. Diening, P. Hästö, The maximal operator on weighted variable Lebesgue spaces, Fractional Calculus Appl. Anal., 14 (2011), 361-374. doi: 10.2478/s13540-011-0023-7
[17]	D. Cruz-Uribe, A. Fiorenza, C. J. Neugebauer, Weighted norm inequalities for the maximal operator on variable Lebesgue spaces, J. Math. Anal. Appl., 394 (2012), 744-760. doi: 10.1016/j.jmaa.2012.04.044
[18]	A. Lerner, On a dual property of the maximal operator on weighted variable Lp spaces, Contemp. Math., 693 (2017), 283-300. doi: 10.1090/conm/693/13932
[19]	L. Diening, P. Hästö, Muckenhoupt weights in variable exponent spaces, 2008. Available from: https://www.researchgate.net/publication/228779582.
[20]	M. Izuki, T. Noi, An intrinsic square function on weighted Herz spaces with variable exponent, 2016. Available from: https://arXiv.org/abs/1606.01019.
[21]	B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Am. Math. Soc., 165 (1972), 207-226. doi: 10.1090/S0002-9947-1972-0293384-6
[22]	M. Izuki, Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent, Rend. Circ. Mat. Palermo, 59 (2010), 461-472. doi: 10.1007/s12215-010-0034-y
[23]	M. Izuki, Boundedness of commutators on Herz spaces with variable exponent, Rend. Circ. Mat. Palermo, 59 (2010), 199-213. doi: 10.1007/s12215-010-0015-1
[24]	M. Izuki, Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J., 40 (2010), 343-355.

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