# Article

Full entry | PDF   (0.2 MB)
Keywords:
commutator; BMO; fractional maximal function; variable exponent Lebesgue space
Summary:
Let $M_{\beta }$ be the fractional maximal function. The commutator generated by $M_{\beta }$ and a suitable function $b$ is defined by $[M_{\beta },b]f = M_{\beta }(bf)-bM_{\beta }(f)$. Denote by $\mathscr {P}(\mathbb R^{n})$ the set of all measurable functions $p(\cdot )\colon \mathbb R^{n}\to [1,\infty )$ such that $$1< p_{-}:=\mathop {\rm ess inf}_{x\in \mathbb R^n}p(x) \quad \text {and}\quad p_{+}:=\mathop {\rm ess sup}_{x\in \mathbb R^n}p(x)<\infty ,$$ and by $\mathscr {B}(\mathbb R^{n})$ the set of all $p(\cdot ) \in \mathscr {P}(\mathbb R^{n})$ such that the Hardy-Littlewood maximal function $M$ is bounded on $L^{p(\cdot )}(\mathbb R^{n})$. In this paper, the authors give some characterizations of $b$ for which $[M_{\beta },b]$ is bounded from $L^{p(\cdot )}(\mathbb R ^{n})$ into $L^{q(\cdot )}(\mathbb R^{n})$, when $p(\cdot )\in \mathscr {P}(\mathbb R^{n})$, $0<{\beta }<n/p_{+}$ and $1/q(\cdot )=1/p(\cdot )-\beta /n$ with $q(\cdot )(n-\beta )/n \in \mathscr {B}(\mathbb R^{n})$.
References:
[1] Bastero, J., Milman, M., Ruiz, F. J.: Commutators for the maximal and sharp functions. Proc. Am. Math. Soc. (electronic) 128 (2000), 3329-3334. DOI 10.1090/S0002-9939-00-05763-4 | MR 1777580 | Zbl 0957.42010
[2] 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
[3] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103 (1976), 611-635. MR 0412721 | Zbl 0326.32011
[4] 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
[5] Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.: The maximal function on variable $L^{p}$ spaces. Ann. Acad. Sci. Fenn., Math. 28 (2003), 223-238. MR 1976842
[6] Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129 (2005), 657-700. DOI 10.1016/j.bulsci.2003.10.003 | MR 2166733
[7] 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
[8] Huang, A. W., Xu, J. S.: Multilinear singular integrals and commutators in variable exponent Lebesgue spaces. Appl. Math., Ser. B (Engl. Ed.) 25 (2010), 69-77. MR 2606534 | Zbl 1224.42030
[9] 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
[10] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. MR 1134951
[11] Milman, M., Schonbek, T.: Second order estimates in interpolation theory and applications. Proc. Am. Math. Soc. 110 (1990), 961-969. DOI 10.1090/S0002-9939-1990-1075187-4 | MR 1075187 | Zbl 0717.46066
[12] Segovia, C., Torrea, J. L.: Vector-valued commutators and applications. Indiana Univ. Math. J. 38 (1989), 959-971. MR 1029684 | Zbl 0696.47033
[13] Segovia, C., Torrea, J. L.: Higher order commutators for vector-valued Calderón-Zygmund operators. Trans. Am. Math. Soc. 336 (1993), 537-556. MR 1074151 | Zbl 0799.42009
[14] Xu, J. S.: The boundedness of multilinear commutators of singular integrals on Lebesgue spaces with variable exponent. Czech. Math. J. 57 (2007), 13-27. DOI 10.1007/s10587-007-0040-1 | MR 2309945 | Zbl 1174.42312
[15] Zhang, P., Wu, J. L.: Commutators of the fractional maximal functions. Acta Math. Sin., Chin. Ser. 52 (2009), 1235-1238. MR 2640953 | Zbl 1212.42067

Partner of