## Banach Journal of Mathematical Analysis

### Intrinsic atomic and molecular decompositions of Hardy–Musielak–Orlicz spaces

Kwok-Pun Ho

#### Abstract

We introduce the Hardy type space for Musielak–Orlicz spaces. It includes several existing Hardy type spaces such as the Hardy–Orlicz spaces and the Hardy spaces with variable exponents. Furthermore, we develop an atomic decomposition such that the size condition just relies on the norms of Musielak–Orlicz spaces. This gives us a nature extension of the molecular decompositions to the Hardy type space for Musielak–Orlicz spaces.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 566-592.

Dates
Accepted: 11 November 2015
First available in Project Euclid: 22 July 2016

https://projecteuclid.org/euclid.bjma/1469199410

Digital Object Identifier
doi:10.1215/17358787-3607354

Mathematical Reviews number (MathSciNet)
MR3528348

Zentralblatt MATH identifier
1346.42024

#### Citation

Ho, Kwok-Pun. Intrinsic atomic and molecular decompositions of Hardy–Musielak–Orlicz spaces. Banach J. Math. Anal. 10 (2016), no. 3, 566--592. doi:10.1215/17358787-3607354. https://projecteuclid.org/euclid.bjma/1469199410

#### References

• [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988.
• [2] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645.
• [3] D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Heidelberg, 2013.
• [4] D. Cruz-Uribe, J. Martell, and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Oper. Theory Adv. Appl. 215, Birkhäuser, Basel, 2011.
• [5] L. Diening, P. Harjulehto, P. Hästö, and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011.
• [6] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985.
• [7] K.-P. Ho, Characterization of $\mathit{BMO}$ in terms of rearrangement-invariant Banach function spaces, Expo. Math. 27 (2009), no. 4, 363–372.
• [8] K.-P. Ho, Littlewood–Paley spaces, Math. Scand. 108 (2011), no. 1, 77–102.
• [9] K.-P. Ho, Atomic decompositions of Hardy spaces and characterization of $\mathit{BMO}$ via Banach function spaces, Anal. Math. 38 (2012), no. 3, 173–185.
• [10] K.-P. Ho, Vector-valued singular integral operators on Morrey type spaces and variable Triebel–Lizorkin–Morrey spaces, Ann. Acad. Sci. Fenn. Math. 37 (2012), no. 2, 375–406.
• [11] K.-P. Ho, Atomic decompositions of weighted Hardy–Morrey spaces, Hokkaido Math. J. 42 (2013), no. 1, 131–157.
• [12] K.-P. Ho, Atomic decomposition of Hardy–Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 31–62.
• [13] K.-P. Ho, Vector-valued operators with singular kernel and Triebel–Lizorkin-block spaces with variable exponents, Kyoto J. Math. 56 (2016), no. 1, 97–124.
• [14] K.-P. Ho, Atomic decompositions of weighted Hardy spaces with variable exponents, preprint, to appear in Tohoku Math. J. (2016).
• [15] S. Hou, D. Yang, and S. Yang, Lusin area function and molecular characterizations of Musielak–Orlicz Hardy spaces and their applications, Commun. Contemp. Math. 15 (2013), no. 6, art. ID 1350029.
• [16] H. Jia and H. Wang, Decomposition of Hardy–Morrey spaces, J. Math. Anal. Appl. 354 (2009), no. 1, 99–110.
• [17] R. Jiang and D. Yang, New Orlicz–Hardy spaces associated with divergence form of elliptic operator, J. Funct. Anal. 258 (2010), no. 4, 1167–1224.
• [18] R. Jiang, D. Yang, and Y. Zhou, Orlicz–Hardy spaces associated with operators, Sci. China Ser. A 52 (2009), no. 5, 1042–1080.
• [19] L. D. Ky, New Hardy spaces of Musielak–Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), no. 1, 115–150.
• [20] Y. Liang, J. Huang, and D. Yang, New real-variable characterizations of Musielak–Orlicz Hardy spaces, J. Math. Anal. Appl. 395 (2012), no. 1, 413–428.
• [21] Y. Liang, E. Nakai, D. Yang, and J. Zhang, Boundedness of intrinsic Littlewood–Paley functions on Musielak–Orlicz Morrey and Campanato spaces, Banach J. Math. Anal. 8 (2014), no. 1, 221–268.
• [22] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I. Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin–New York, 1977.
• [23] S. Lu, Four Lectures on Real $H_{p}$ Spaces, World Scientific, River Edge, NJ, 1995.
• [24] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983.
• [25] E. Nakai, Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95–104.
• [26] E. Nakai, Orlicz–Morrey spaces and the Hardy–Littlewood maximal function, Studia Math. 188 (2008), no. 3, 193–221.
• [27] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
• [28] E. Nakai and Y. Sawano, Orlicz–Hardy spaces and their duals, Sci. China Math. 57 (2014), no. 5, 903–962.
• [29] S. Okada, W. Ricker, and E. Sánchez Pérez, Optimal Domain and Integral Extension of Operators, Oper. Theory Adv. Appl. 180, Birkhäuser, Basel, 2008.
• [30] Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), no. 1, 123–148.
• [31] Y. Sawano and H. Tanaka, Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces, Math. Z. 257 (2007), no. 4, 871–905.
• [32] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
• [33] M. Taibleson and G. Weiss, “The molecular characterization of certain Hardy spaces” in Representation Theorems for Hardy Spaces, Astérisque 77, Soc. Math. France, Paris, 67–149.
• [34] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Dover, Mineola, NY, 2004.
• [35] D. Yang and S. Yang, Weighted local Orlicz–Hardy spaces with applications to pseudo-differential operators, Dissertationes Math. (Rozprawy Mat.) 478 (2011), 1–78.
• [36] D. Yang and S. Yang, Local Hardy spaces of Musielak–Orlicz type and their applications, Sci. China Math. 55 (2012), no. 8, 1677–1720.
• [37] W. Yuan, W. Sickel, and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Math. 2005, Springer, Berlin, 2010.