Annals of Functional Analysis

Atomic decomposition of variable Hardy spaces via Littlewood–Paley–Stein theory

Jian Tan

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


The purpose of this paper is to give a new atomic decomposition for variable Hardy spaces via the discrete Littlewood–Paley–Stein theory. As an application of this decomposition, we assume that T is a linear operator bounded on Lq and Hp(), and we thus obtain that T can be extended to a bounded operator from Hp() to Lp().

Article information

Ann. Funct. Anal., Volume 9, Number 1 (2018), 87-100.

Received: 21 September 2016
Accepted: 23 February 2017
First available in Project Euclid: 14 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 42B30: $H^p$-spaces

variable exponent atomic decomposition Hardy spaces Littlewood–Paley–Stein functions


Tan, Jian. Atomic decomposition of variable Hardy spaces via Littlewood–Paley–Stein theory. Ann. Funct. Anal. 9 (2018), no. 1, 87--100. doi:10.1215/20088752-2017-0026.

Export citation


  • [1] R. Coifman, A real variable characterization of $H^{p}$, Studia Math. 51 (1974), 269–274.
  • [2] R. Coifman and Y. Meyer, Wavelets, Calderón–Zygmund and multilinear operators, Cambridge Stud. Adv. Math. 48, Cambridge Univ. Press, Cambridge, 1992.
  • [3] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser, Basel, 2013.
  • [4] 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), no. 1, 239–264.
  • [5] D. Cruz-Uribe and L. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493.
  • [6] D. Deng and Y.-S. Han, Harmonic Analysis on Spaces of Homogeneous Type, Lecture Notes in Math. 1966, Springer, Berlin, 2009.
  • [7] L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657–700.
  • [8] L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Heidelberg, 2011.
  • [9] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), no. 4, 777–799.
  • [10] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170.
  • [11] Y.-S. Han, M-Y. Lee, and C.-C. Lin, Atomic decomposition and boundedness of operators on weighted Hardy spaces, Canad. Math. Bull. 55 (2012), no. 2, 303–314.
  • [12] Y.-S. Han and E. T. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 110 (1994), no. 530.
  • [13] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41 (1991), no. 4, 592–618.
  • [14] R. Latter, A characterization of $H^{p}(\mathbb{R}^{n})$ in terms of atoms, Studia Math. 62 (1978), no. 1, 93–101.
  • [15] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992.
  • [16] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
  • [17] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950.
  • [18] W. Orlicz, Über konjugierte Exponentenfolgen, Stud. Math. 3 (1931), 200–211.
  • [19] L. Pick and M. Růžička, 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.
  • [20] K. Zhao and Y.-S. Han, Boundedness of operators on Hardy spaces, Taiwanese J. Math. 14 (2010), no. 2, 319–327.
  • [21] C. Zhuo, D. Yang, and Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents, Bull. Malays. Math. Sci. Soc. (2) 39 (2016), no. 4, 1541–1577.