Journal of the Mathematical Society of Japan

A note on maximal commutators and commutators of maximal functions


Full-text: Open access


In this paper maximal commutators and commutators of maximal functions with functions of bounded mean oscillation are investigated. New pointwise estimates for these operators are proved.

Article information

J. Math. Soc. Japan, Volume 67, Number 2 (2015), 581-593.

First available in Project Euclid: 21 April 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis

maximal operator commutator BMO


AGCAYAZI, Mujdat; GOGATISHVILI, Amiran; KOCA, Kerim; MUSTAFAYEV, Rza. A note on maximal commutators and commutators of maximal functions. J. Math. Soc. Japan 67 (2015), no. 2, 581--593. doi:10.2969/jmsj/06720581.

Export citation


  • A. M. Alphonse, An end point estimate for maximal commutators, J. Fourier Anal. Appl., 6 (2000), 449–456.
  • J. Bastero, M. Milman and Francisco J. Ruiz, Commutators for the maximal and sharp functions, Proc. Amer. Math. Soc., 128 (2000), 3329–3334 (electronic).
  • V. Bennett and R. Sharpley, Weak-type inequalities for $H^{p}$ and BMO, In: Harmonic Analysis in Euclidean Spaces. Part 1, Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978, (eds. G. Weiss and S. Wainger), Proc. Sympos. Pure Math., 35, part 2, Amer. Math. Soc., Providence, RI, 1979, pp.,201–229.
  • A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in BMO and $H^1$, Ann. Inst. Fourier (Grenoble), 57 (2007), 1405–1439.
  • C. Bennett and R. Sharpley, R., Interpolation of Operators, Pure Appl. Math., 129, Academic Press Inc., Boston, MA, 1988.
  • R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2), 103 (1976), 611–635.
  • J. García-Cuerva, J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud., 116, Notas Mat., 104, North-Holland Publishing Co., Amsterdam, 1985.
  • J. García-Cuerva, E. Harboure, C. Segovia and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J., 40 (1991), 1397–1420.
  • L. Grafakos, Modern Fourier Analysis. 2nd ed., Grad. Texts in Math., 250, Springer-Verlag, New York, 2009.
  • G. Hu, H. Lin and D. Yang, Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type, Abstr. Appl. Anal., 2008, (2008), Art. ID 237937.
  • G. Hu and D. Yang, Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type, J. Math. Anal. Appl., 354 (2009), 249–262.
  • S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat., 16 (1978), 263–270.
  • F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415–426.
  • D. Li, G. Hu and X. Shi, Weighted norm inequalities for the maximal commutators of singular integral operators, J. Math. Anal. Appl., 319 (2006), 509–521.
  • M. Milman and T. Schonbek, Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc., 110 (1990), 961–969.
  • C. Pérez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128 (1995), 163–185.
  • M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker Inc., New York, 1991.
  • C. Segovia and J. L. Torrea, Weighted inequalities for commutators of fractional and singular integrals, In: Conference on Mathematical Analysis, El Escorial, 1989, Publ. Mat., 35, Universitat Autònoma de Barcelona, Departament de Matemátiques, Barcelona, 1991, pp.,209–235.
  • C. Segovia and J. L. Torrea, Higher order commutators for vector-valued Calderón-Zygmund operators, Trans. Amer. Math. Soc., 336 (1993), 537–556.