## Annals of Functional Analysis

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

Jian Tan

#### Abstract

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 $L^{q}$ and $H^{p(\cdot)}$, and we thus obtain that $T$ can be extended to a bounded operator from $H^{p(\cdot)}$ to $L^{p(\cdot)}$.

#### Article information

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

Dates
Accepted: 23 February 2017
First available in Project Euclid: 14 August 2017

https://projecteuclid.org/euclid.afa/1502697622

Digital Object Identifier
doi:10.1215/20088752-2017-0026

Mathematical Reviews number (MathSciNet)
MR3758745

Zentralblatt MATH identifier
06841343

#### Citation

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. https://projecteuclid.org/euclid.afa/1502697622

#### References

• [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.