Tokyo Journal of Mathematics

John-Nirenberg Inequalities with Variable Exponents on Probability Spaces

Zhiwei HAO, Yong JIAO, and Lian WU

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we study the John-Nirenberg inequalities with variable exponents on a probability space. Let $Y$ be a rearrangement invariant Banach function space defined on $(\Omega,\mathcal{F},P)$ and a measurable function $p(\cdot): \Omega\rightarrow \mathbf{R}^+$ be a variable exponent. We prove that if the stochastic basis is regular, then $$BMO_{\phi,Y}=BMO_{\phi,p(\cdot)}\,,\quad \forall 1\leq p(\cdot)<\infty\,,$$ where $\phi(r)=1/r\Phi^{-1}(1/r)$ and $\Phi$ is a concave function with proper condition.

Article information

Tokyo J. Math., Volume 38, Number 2 (2015), 353-367.

First available in Project Euclid: 14 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter
Secondary: 60G46: Martingales and classical analysis


WU, Lian; HAO, Zhiwei; JIAO, Yong. John-Nirenberg Inequalities with Variable Exponents on Probability Spaces. Tokyo J. Math. 38 (2015), no. 2, 353--367. doi:10.3836/tjm/1452806045.

Export citation


  • A. Almeida and P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), 1628–1655.
  • C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, New York, 1988.
  • D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser, 2013.
  • G. Cuerva, Weighted norm inequalities and related topics, North-Holland, Amsterdam, 1985.
  • D. Cruz-Uribe, SFO and L. Daniel Wang, Variable Hardy spaces, to appear in Indiana University Mathematics Journal, arXiv:1211.6505v1, 2014.
  • L. Diening and P. Harjulehto, Lebesgue and Sololev spaces with variable exponent, Springer, 2011.
  • L. Diening, Maximal functions on generalized $L^{p(\cdot)}$ spaces, Math. Inequal. Appl. 7(2) (2004), 245–253.
  • L. Diening, P. Hästö and R. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), 1731–1768.
  • D. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245–1254.
  • X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)},$ J. Math. Anal. Appl. 263 (2001), 424–446.
  • A. M. Garsia, Martingale inequalities: Seminar notes on recent progress, Math. Lecture Notes Series, New York: Benjamin Inc., 1973.
  • C. Herz, Bounded mean oscillation and regulated martingales, Trans. Amer. Math. Soc. 193 (1974), 199–215.
  • P. Harjulehto, P. Hästö and M. Pere, Variable exponent Lebesgue spaces on metric spaces: The Hardy-Littlewood maximal operator, Real Anal. Exchange 30(1) (2004), 87–104.
  • R. L. Long, Martingale Spaces and Inequalities, Peking University Press, Beijing, 1993.
  • Y. Jiao, Carleson measures and vector-valued BMO martingales, Probab. Theory Relat. Fields 145 (2009), 421–434.
  • Y. Jiao, L. Wu and M. Popa, Operator-valued martingale transforms in rearrangement invariant spaces and applications, Sci. China Math. 56(4) (2013), 831–844.
  • Y. Jiao, D. Zhou, Z. Hao and W. Chen, Martingale Hardy spaces with variable exponents, arXiv:1404.2395v2, 2014.
  • F. John and L. Nirenberg, On the functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.
  • W.B. Johnson and G. Schechtman, Martingale inequalities in rearrangement invariant function spaces, Israel J. Math. 64(3) (1998), 267–275.
  • M. Izuki and Y. Sawano, Variable Lebesgue norm estimates for BMO functions, Czechoslovak Math. J. 62 (137) (2012), 717–727.
  • M. Izuki, Y. Sawano and Y. Tsutsui, Variable Lebesgue norm estimates for BMO functions II preprint, 2013.
  • K. P. Ho, John-Nirenberg inequalities on Lebesgue spaces with variable exponents, Taiwanese J. Math. 18(4) (2014), 1107–1118.
  • M. Kikuchi, New martingale inequalities in rearrangement function spaces, Proc. Edinb. Math. Soc. 47(2) (2004), 633–657.
  • S. Krein, Y. Petunin and E. Semenov, Interpolation of linear operators, Amer. Math. Soc., Providence, 1982.
  • O. Kovàčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math. J. 41(116) (1991), 592–618.
  • T. Miyamoto, E. Nakai and G. Sadasue, Martingale Orlicz-Hardy spaces, Math. Nachr. 285 (2012), 670–686.
  • E. Nakai and G. Sadasue, Martingale Morrey-Campanato spaces and fractional integrals, J. Funct. Spaces Appl. (2012) Article ID673929, 29 pp.
  • N. I. Ya, Martingale inequalities in rearrangement invariant function spaces, In Function spaces, Teubner-Texte Math., Teubnet, Stuttgart, 120 (1991), 120–127.
  • E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), 3665–3748.
  • W. Orlicz, Über konjugierte Expoentenfolgen, Studia Math. 3 (1931), 200–211.
  • F. Weisz, Martingale Hardy spaces and their applications in Fourier analysis, Lecture Notes in Mathematics, 1568, Springer-Verlag, Berlin, 1994.
  • R. Yi, L. Wu and Y. Jiao, New John-Nirenberg inequalities for martingales, Statist. Probab. Lett. 86 (2014), 68–73.