Geodesic
Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 251-269
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-016-0254-1
Classification : 42B20, 42B35
Keywords: Herz space; variable exponent; commutator; Marcinkiewicz integral 
@article{10_1007_s10587_016_0254_1,
     author = {Wang, Hongbin},
     title = {Commutators of {Marcinkiewicz} integrals on {Herz} spaces with variable exponent},
     journal = {Czechoslovak Mathematical Journal},
     pages = {251--269},
     year = {2016},
     volume = {66},
     number = {1},
     doi = {10.1007/s10587-016-0254-1},
     mrnumber = {3483237},
     zbl = {06587888},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0254-1/}
}
TY  - JOUR
AU  - Wang, Hongbin
TI  - Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent
JO  - Czechoslovak Mathematical Journal
PY  - 2016
SP  - 251
EP  - 269
VL  - 66
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0254-1/
DO  - 10.1007/s10587-016-0254-1
LA  - en
ID  - 10_1007_s10587_016_0254_1
ER  - 
%0 Journal Article
%A Wang, Hongbin
%T Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent
%J Czechoslovak Mathematical Journal
%D 2016
%P 251-269
%V 66
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0254-1/
%R 10.1007/s10587-016-0254-1
%G en
%F 10_1007_s10587_016_0254_1
Wang, Hongbin. Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 251-269. doi: 10.1007/s10587-016-0254-1

[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 | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | MR

[6] Ding, Y., Lu, S., Yabuta, K.: On commutators of Marcinkiewicz integrals with rough kernel. J. Math. Anal. Appl. 275 (2002), 60-68. | DOI | MR | Zbl

[7] Izuki, M.: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo (2) 59 (2010), 199-213. | DOI | MR | Zbl

[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 | MR | Zbl

[9] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. | MR

[10] Liu, Z., Wang, H.: Boundedness of Marcinkiewicz integrals on Herz spaces with variable exponent. Jordan J. Math. Stat. 5 (2012), 223-239. | Zbl

[11] Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161 (1971), 249-258. | DOI | MR

[12] Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262 (2012), 3665-3748. | DOI | MR | Zbl

[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 | MR

[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 | Zbl

[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 | Zbl

Cité par Sources :