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MSC: 42B20, 42B35
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Herz space; variable exponent; commutator; Marcinkiewicz integral
Let $\Omega \in L^s({\mathrm S}^{n-1})$ for $s\geq 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral $\mu _\Omega $ and $b$ is defined by \begin {equation*} \displaystyle [b,\mu _\Omega ] (f)(x)=\biggl (\int ^\infty _0\biggl |\int _{|x-y|\leq t} \frac {\Omega (x-y)}{|x-y|^{n-1}}[b(x)-b(y)]f(y) {\rm d} y\bigg |^2\frac {{\rm d} t}{t^3}\bigg )^{1/2}. \end {equation*} In this paper, the author proves the $(L^{p(\cdot )}(\mathbb {R}^{n}),L^{p(\cdot )}(\mathbb {R}^{n}))$-boundedness of the Marcinkiewicz integral operator $\mu _\Omega $ and its commutator $[b,\mu _\Omega ]$ when $p(\cdot )$ satisfies some conditions. Moreover, the author obtains the corresponding result about $\mu _\Omega $ and $[b,\mu _\Omega ]$ on Herz spaces with variable exponent.
[1] Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator and fractional integrals on variable $L^p$ spaces. Rev. Mat. Iberoam. 23 (2007), 743-770. DOI 10.4171/RMI/511 | MR 2414490
[2] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis Birkhäuser/Springer, New York (2013). MR 3026953
[3] Cruz-Uribe, D., Fiorenza, A., Martell, J. M., Pérez, C.: The boundedness of classical operators on variable $L^p$ spaces. Ann. Acad. Sci. Fenn. Math. 31 (2006), 239-264. MR 2210118 | Zbl 1100.42012
[4] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017 Springer, Berlin (2011). MR 2790542 | Zbl 1222.46002
[5] Ding, Y., Fan, D., Pan, Y.: Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana Univ. Math. J. 48 (1999), 1037-1055. DOI 10.1512/iumj.1999.48.1696 | MR 1736970
[6] Ding, Y., Lu, S., Yabuta, K.: On commutators of Marcinkiewicz integrals with rough kernel. J. Math. Anal. Appl. 275 (2002), 60-68. DOI 10.1016/S0022-247X(02)00230-5 | MR 1941772
[7] Izuki, M.: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo (2) 59 (2010), 199-213. DOI 10.1007/s12215-010-0015-1 | MR 2670690 | Zbl 1202.42029
[8] Izuki, M.: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 36 (2010), 33-50. DOI 10.1007/s10476-010-0102-8 | MR 2606575 | Zbl 1224.42025
[9] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. MR 1134951
[10] Liu, Z., Wang, H.: Boundedness of Marcinkiewicz integrals on Herz spaces with variable exponent. Jordan J. Math. Stat. 5 (2012), 223-239.
[11] Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161 (1971), 249-258. DOI 10.1090/S0002-9947-1971-0285938-7 | MR 0285938
[12] Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262 (2012), 3665-3748. DOI 10.1016/j.jfa.2012.01.004 | MR 2899976 | Zbl 1244.42012
[13] Stein, E. M.: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Am. Math. Soc. 88 (1958), 430-466 corr. ibid. 98 186 (1961). DOI 10.1090/S0002-9947-1958-0112932-2 | MR 0112932
[14] Tan, J., Liu, Z. G.: Some boundedness of homogeneous fractional integrals on variable exponent function spaces. Acta Math. Sin., Chin. Ser. 58 (2015), 309-320 Chinese. MR 3408398
[15] Wang, H., Fu, Z., Liu, Z.: Higher-order commutators of Marcinkiewicz integrals on variable Lebesgue spaces. Acta Math. Sci., Ser. A Chin. Ed. 32 (2012), 1092-1101 Chinese. MR 3075205
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