Banach Journal of Mathematical Analysis

Martingale Hardy spaces with variable exponents

Yong Jiao, Dejian Zhou, Zhiwei Hao, and Wei Chen

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In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a $(1,p(\cdot),\infty)$-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of $p(\cdot)$ to replace the so-called log-Hölder continuity condition in $\mathbb{R}^{n}$.

Article information

Banach J. Math. Anal. Volume 10, Number 4 (2016), 750-770.

Received: 7 October 2015
Accepted: 11 January 2016
First available in Project Euclid: 20 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

martingale Hardy spaces variable exponents atomic decomposition


Jiao, Yong; Zhou, Dejian; Hao, Zhiwei; Chen, Wei. Martingale Hardy spaces with variable exponents. Banach J. Math. Anal. 10 (2016), no. 4, 750--770. doi:10.1215/17358787-3649326.

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  • [1] A. Almeida and P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), no. 5, 1628–1655.
  • [2] H. Aoyama, Lebesgue spaces with variable exponent on a probability space, Hiroshima Math. J. 39 (2009), no. 2, 207–216.
  • [3] D. Cruz-Uribe SFO and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser, Heidelberg, 2013.
  • [4] D. Cruz-Uribe SFO, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239–264.
  • [5] D. Cruz-Uribe SFO and L. Daniel Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493.
  • [6] L. Diening, Maximal functions on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004), no. 2, 245–253.
  • [7] L. Diening, P. Harjulehto, P. Hästö, and M. Ru\u{z}ička, Lebesgue and Sololev Spaces with Variable Exponent, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011.
  • [8] L. Diening, P. Hästö, and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), no. 6, 1731–1768.
  • [9] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446.
  • [10] A. M. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, Benjamin, 1973.
  • [11] Z. Hao and Y. Jiao, Fractional integral on martingale Hardy spaces with variable exponents, Fract. Calc. Appl. Anal. 18 (2015), no. 5, 1128–1145.
  • [12] P. Harjulehto, P. Hästö, and M. Pere, Variable exponent Lebesgue spaces on metric spaces: The Hardy-Littlewood maximal operator, Real Anal. Exchange 30 (2004), no. 1, 87–103.
  • [13] K. Ho, John-Nirenberg inequalities on Lebesgue spaces with variable exponents, Taiwanese J. Math. 18 (2014), no. 4, 1107–1118.
  • [14] K. Ho, Atomic decomposition of Hardy-Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 31–62.
  • [15] K. Ho, Vector-valued John–Nirenberg inequalities and vector-valued mean oscillations characterization of $BMO$, Results. Math., published electronically July 14, 2015.
  • [16] M. Izuki, Y. Sawan and Y. Tsutsui, Variable Lebesgue norm estimates for BMO functions, II, Anal. Math. 40 (2014), no. 3, 215–230.
  • [17] O. Kovàčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1,p(x)}$, Czechoslovak Math. J. 41 (1991), no. 4, 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] P. Liu and M. Wang, Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent, preprint, arXiv:1412.8146v1.
  • [20] R. Long, Martingale Spaces and Inequalities, Peking Univ. Press, Beijing, 1993.
  • [21] E. Nakai and G. Sadasue, Maximal function on generalized martingale Lebesgue spaces with variable exponent, Statist. Probab. Lett. 83 (2013), no. 10, 2168–2171.
  • [22] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
  • [23] A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(\mathbb{R})$, Math. Inequal. Appl. 7 (2004), no. 2, 255–265.
  • [24] W. Orlicz, Über konjugierte Expoentenfolgen, Studia Math. 3 (1931), 200–211.
  • [25] L. Pick and M. Ruzicka, An example of a space $L^{p(x)}$ on which the Hardy-Littlewood maximal operator is not bounded, Expo. Math. 19 (2001), no. 4, 369–371.
  • [26] Y. Sawano, Atomic decomposition of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), no. 1, 123–148.
  • [27] F. Weisz, Martingale Hardy spaces for $0<p\leq1$, Probab. Theory Related Fields 84 (1990), no. 3, 361–376.
  • [28] F. Weisz, Martingale Hardy spaces and Their Applications in Fourier Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [29] L. Wu, Z. Hao, and Y. Jiao, John-Nirenberg inequalities with variable exponents on probability spaces, Tokyo J. Math. 38 (2015), no. 2, 353–367.
  • [30] D. Yang, C. Zhuo, and W. Yuan, Triebel-Lizorkin type spaces with variable exponents, Banach J. Math. Anal. 9 (2015), no. 4, 146–202.
  • [31] R. Yi, L. Wu, and Y. Jiao, New John-Nirenberg inequalities for martingales, Statist. Probab. Lett. 86 (2014), 68–73.