| Title:
|
Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent (English) |
| Author:
|
Wang, Hongbin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
251-269 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Herz space |
| Keyword:
|
variable exponent |
| Keyword:
|
commutator |
| Keyword:
|
Marcinkiewicz integral |
| MSC:
|
42B20 |
| MSC:
|
42B35 |
| idZBL:
|
Zbl 06587888 |
| idMR:
|
MR3483237 |
| DOI:
|
10.1007/s10587-016-0254-1 |
| . |
| Date available:
|
2016-04-07T15:11:43Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144871 |
| . |
| Reference:
|
[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. Zbl 1213.42063, MR 2414490, 10.4171/RMI/511 |
| Reference:
|
[2] Cruz-Uribe, D. V., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis.Applied and Numerical Harmonic Analysis Birkhäuser/Springer, New York (2013). Zbl 1268.46002, MR 3026953 |
| Reference:
|
[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. Zbl 1100.42012, MR 2210118 |
| Reference:
|
[4] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents.Lecture Notes in Mathematics 2017 Springer, Berlin (2011). Zbl 1222.46002, MR 2790542 |
| Reference:
|
[5] Ding, Y., Fan, D., Pan, Y.: Weighted boundedness for a class of rough Marcinkiewicz integrals.Indiana Univ. Math. J. 48 (1999), 1037-1055. MR 1736970, 10.1512/iumj.1999.48.1696 |
| Reference:
|
[6] Ding, Y., Lu, S., Yabuta, K.: On commutators of Marcinkiewicz integrals with rough kernel.J. Math. Anal. Appl. 275 (2002), 60-68. Zbl 1019.42009, MR 1941772, 10.1016/S0022-247X(02)00230-5 |
| Reference:
|
[7] Izuki, M.: Boundedness of commutators on Herz spaces with variable exponent.Rend. Circ. Mat. Palermo (2) 59 (2010), 199-213. Zbl 1202.42029, MR 2670690, 10.1007/s12215-010-0015-1 |
| Reference:
|
[8] Izuki, M.: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization.Anal. Math. 36 (2010), 33-50. Zbl 1224.42025, MR 2606575, 10.1007/s10476-010-0102-8 |
| Reference:
|
[9] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$.Czech. Math. J. 41 (1991), 592-618. MR 1134951 |
| Reference:
|
[10] Liu, Z., Wang, H.: Boundedness of Marcinkiewicz integrals on Herz spaces with variable exponent.Jordan J. Math. Stat. 5 (2012), 223-239. Zbl 1277.42018 |
| Reference:
|
[11] Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for singular and fractional integrals.Trans. Am. Math. Soc. 161 (1971), 249-258. MR 0285938, 10.1090/S0002-9947-1971-0285938-7 |
| Reference:
|
[12] Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces.J. Funct. Anal. 262 (2012), 3665-3748. Zbl 1244.42012, MR 2899976, 10.1016/j.jfa.2012.01.004 |
| Reference:
|
[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). MR 0112932, 10.1090/S0002-9947-1958-0112932-2 |
| Reference:
|
[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. Zbl 1340.42055, MR 3408398 |
| Reference:
|
[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. Zbl 1289.42056, MR 3075205 |
| . |
