Czechoslovak Mathematical Journal, Vol. 69, No. 4, pp. 1029-1037, 2019


Notes on commutator on the variable exponent Lebesgue spaces

Dinghuai Wang

Received December 28, 2017.   Published online March 22, 2019.

Abstract:  We obtain the factorization theorem for Hardy space via the variable exponent Lebesgue spaces. As an application, it is proved that if the commutator of Coifman, Rochberg and Weiss $[b,T]$ is bounded on the variable exponent Lebesgue spaces, then $b$ is a bounded mean oscillation (BMO) function.
Keywords:  bounded mean oscillation; commutator; Hardy space; variable exponent Lebesgue space
Classification MSC:  42B20, 47B07


References:
[1] C. Capone, D. Cruz-Uribe, A. Fiorenza: 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 | Zbl 1213.42063
[2] L. Chaffee, D. Cruz-Uribe: Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces. Math. Inequal. Appl. 21 (2018), 1-16. DOI 10.7153/mia-2018-21-01 | MR 3716205 | Zbl 1384.42012
[3] R. R. Coifman, R. Rochberg, G. Weiss: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 (1976), 611-635. DOI 10.2307/1970954 | MR 0412721 | Zbl 0326.32011
[4] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, C. Pérez: The boundedness of classical operators on variable $L^p$ spaces. Ann. Acad. Sci. Fenn., Math. 31 (2006), 239-264. MR 2210118 | Zbl 1100.42012
[5] L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017 Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[6] S. Janson: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16 (1978), 263-270. DOI 10.1007/BF02386000 | MR 0524754 | Zbl 0404.42013
[7] Y. Komori: Notes on commutators on Herz-type spaces. Arch. Math. 81 (2003), 318-326. DOI 10.1007/s00013-003-0545-2 | MR 2013263 | Zbl 1053.42020
[8] O. Kováčik, J. Rákosník: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. MR 1134951 | Zbl 0784.46029
[9] J. Li, B. D. Wick: Weak factorizations of the Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. Canad. Math. Bull. 60 (2017), 571-585. DOI 10.4153/CMB-2017-033-9 | MR 3679731 | Zbl 1372.42018
[10] A. Uchiyama: The factorization of $H^p$ on the space of homogeneous type. Pac. J. Math. 92 (1981), 453-468. DOI 10.2140/pjm.1981.92.453 | MR 0618077 | Zbl 0493.42032

Affiliations:   Dinghuai Wang, Xinjiang University, 14 Sheng Li Road, Urumqi 830046, P. R. China, e-mail: Wangdh1990@126.com


 
PDF available at: