Kyoto Journal of Mathematics

Boundedness for commutators of fractional integrals on Herz–Morrey spaces with variable exponent

Jianglong Wu

Full-text: Open access

Abstract

In this paper, some boundedness for commutators of fractional integrals is obtained on Herz–Morrey spaces with variable exponent applying some properties of variable exponent and bounded mean oscillation (BMO) functions.

Article information

Source
Kyoto J. Math., Volume 54, Number 3 (2014), 483-495.

Dates
First available in Project Euclid: 14 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1408020873

Digital Object Identifier
doi:10.1215/21562261-2693397

Mathematical Reviews number (MathSciNet)
MR3263547

Zentralblatt MATH identifier
1310.42009

Subjects
Primary: 47B47: Commutators, derivations, elementary operators, etc. 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis

Citation

Wu, Jianglong. Boundedness for commutators of fractional integrals on Herz–Morrey spaces with variable exponent. Kyoto J. Math. 54 (2014), no. 3, 483--495. doi:10.1215/21562261-2693397. https://projecteuclid.org/euclid.kjm/1408020873


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References

  • [1] A. Almeida and D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl. 394 (2012), 781–795.
  • [2] C. Capone, D. Cruz-Uribe, and A. Fiorenza, The fractional maximal operator and fractional integrals on variable $L^{p}$ spaces, Rev. Mat. Iberoam. 23 (2007), 743–770.
  • [3] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7–16.
  • [4] D. Cruz-Uribe, L. Diening, and A. Fiorenza, A new proof of the boundedness of maximal operators on variable Lebesgue spaces, Boll. Unione Mat. Ital. (9) 2 (2009), 151–173.
  • [5] D. Cruz-Uribe, A. Fiorenza, J. Martell, and C. Pérez, The boundedness of classical operators on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), 239–264.
  • [6] D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer, The maximal function on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), 223–238.
  • [7] L. Diening, Maximal functions on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004), 245–253.
  • [8] L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue spaces and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$, Math. Nachr. 268 (2004), 31–43.
  • [9] L. Diening, Maximal functions on Musielak–Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), 657–700.
  • [10] L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta, and T. Shimomura, Maximal functions in variable exponent spaces: Limiting cases of the exponent, Ann. Acad. Sci. Fenn. Math. 34 (2009), 503–522.
  • [11] E. Hernández and D. Yang, Interpolation of Herz spaces and applications, Math. Nachr. 205 (1999), 69–87.
  • [12] M. Izuki, Boundedness of vector-valued sublinear operators on Herz–Morrey spaces with variable exponent, Math. Sci. Res. J. 13 (2009), 243–253.
  • [13] M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math. 36 (2010), 33–50.
  • [14] M. Izuki, Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent, Rend. Circ. Mat. Palermo (2) 59 (2010), 461–472.
  • [15] M. Izuki, Fractional integrals on Herz–Morrey spaces with variable exponent, Hiroshima Math. J. 40 (2010), 343–355.
  • [16] T. Kopaliani, Infimal convolution and Muckenhoupt $A_{p(\cdot)}$ condition in variable $L^{p}$ spaces, Arch. Math. (Basel) 89 (2007), 185–192.
  • [17] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. 41 (1991), 592–618.
  • [18] A. Lerner, On some questions related to the maximal operator on variable $L^{p}$ spaces, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4229–4242.
  • [19] X. Li and D. Yang, Boundedness of some sublinear operators on Herz spaces, Illinois J. Math. 40 (1996), 484–501.
  • [20] S. Lu and L. Xu, Boundedness of rough singular integral operators on the homogeneous Herz–Morrey spaces, Hokkaido Math. J. 34 (2005), 299–314.
  • [21] S. Lu and D. Yang, The decomposition of the weighted Herz spaces and its application, Sci. China Ser. A 38 (1995), 147–158.
  • [22] A. Nekvinda, Hardy–Littlewood maximal operator on $L^{p(x)}(\mathbb{R}^{n})$, Math. Inequal. Appl. 7 (2004), 255–265.
  • [23] L. Pick and M. Ruz̆ička, An example of a space $L^{p(\cdot)}$ on which the Hardy–Littlewood maximal operator is not bounded, Expo. Math. 19 (2001), 369–371.