Banach Journal of Mathematical Analysis

Wavelet characterizations of Musielak–Orlicz Hardy spaces

Xing Fu and Dachun Yang

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 article, via establishing a new atomic characterization of the Musielak–Orlicz Hardy space Hφ(Rn) [which is essentially deduced from the known molecular characterization of Hφ(Rn)] and some estimates on a new discrete Littlewood–Paley g-function and a Peetre-type maximal function, together with using the known intrinsic g-function characterization of Hφ(Rn), the authors obtain several equivalent characterizations of Hφ(Rn) in terms of wavelets, which extend the wavelet characterizations of both Orlicz–Hardy spaces and the weighted Hardy spaces, and are available to the typical and useful Musielak–Orlicz Hardy space Hlog(Rn). The novelty of this approach is that the new adapted atomic characterization of Hφ(Rn) compensates the inconvenience in applications of the supremum appearing in the original definition of atoms, which play crucial roles in the proof of the main theorem of this article and may have further potential applications.

Article information

Banach J. Math. Anal., Volume 12, Number 4 (2018), 1017-1046.

Received: 29 November 2017
Accepted: 26 March 2018
First available in Project Euclid: 4 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C40: Wavelets and other special systems
Secondary: 42B30: $H^p$-spaces 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory

Musielak–Orlicz Hardy space wavelet atom Peetre-type maximal function Littlewood–Paley g-function


Fu, Xing; Yang, Dachun. Wavelet characterizations of Musielak–Orlicz Hardy spaces. Banach J. Math. Anal. 12 (2018), no. 4, 1017--1046. doi:10.1215/17358787-2018-0010.

Export citation


  • [1] A. Bonami, J. Feuto, and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat. 54 (2010), no. 2, 341–358.
  • [2] A. Bonami and S. Grellier, Hankel operators and weak factorization for Hardy-Orlicz spaces, Colloq. Math. 118 (2010), no. 1, 107–132.
  • [3] A. Bonami, S. Grellier, and L. D. Ky, Paraproducts and products of functions in $BMO({\mathbb{R}}^{n})$ and ${\mathcal{H}}^{1}({\mathbb{R}}^{n})$ through wavelets, J. Math. Pures Appl. (9) 97 (2012), no. 3, 230–241.
  • [4] A. Bonami, T. Iwaniec, P. Jones, and M. Zinsmeister, On the product of functions in BMO and $H^{1}$, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1405–1439.
  • [5] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996.
  • [6] C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193.
  • [7] J. García-Cuerva and J. M. Martell, Wavelet characterization of weighted spaces, J. Geom. Anal. 11 (2001), no. 2, 241–264.
  • [8] E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996.
  • [9] E. Hernández, G. Weiss, and D. Yang, The $\varphi $-transform and wavelet characterizations of Herz-type spaces, Collect. Math. 47 (1996), no. 3, 285–320.
  • [10] 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.
  • [11] M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math. 36 (2010), no. 2, 33–50.
  • [12] M. Izuki, E. Nakai, and Y. Sawano, Wavelet characterization and modular inequalities for weighted Lebesgue spaces with variable exponent, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 551–571.
  • [13] T. S. Kopaliani, Greediness of the wavelet system in $L^{p(t)}({\mathbb{R}})$ spaces, East J. Approx. 14 (2008), no. 1, 59–67.
  • [14] L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2931–2958.
  • [15] L. D. Ky, Bilinear decompositions for the product space $H^{1}_{L}\times\mathrm{BMO}_{L}$, Math. Nachr. 287 (2014), no. 11–12, 1288–1297.
  • [16] 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.
  • [17] L. D. Ky, Endpoint estimates for commutators of singular integrals related to Schrödinger operators, Rev. Mat. Iberoam. 31 (2015), no. 4, 1333–1373.
  • [18] 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.
  • [19] 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.
  • [20] Y. Liang and D. Yang, Musielak-Orlicz Campanato spaces and applications, J. Math. Anal. Appl. 406 (2013), no. 1, 307–322.
  • [21] Y. Liang and D. Yang, Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3225–3256.
  • [22] H. P. Liu, The wavelet characterization of the space weak $H^{1}$, Studia Math. 103 (1992), no. 1, 109–117.
  • [23] S. Lu, Four Lectures on Real $H^{p}$ Spaces, World Scientific, River Edge, NJ, 1995.
  • [24] Y. Meyer, Wavelets and Operators, Cambridge Stud. Adv. Math. 37, Cambridge Univ. Press, Cambridge, 1992.
  • [25] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.
  • [26] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton Math. Ser. 43, Princeton, 1993.
  • [27] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables, I: The theory of $H^{p}$-spaces, Acta Math. 103 (1960), 25–62.
  • [28] M. H. Taibleson and G. Weiss, “The molecular characterization of certain Hardy spaces” in Representation Theorems for Hardy Spaces, Astérisque 77, Soc. Math. France, Paris, 1980, 67–149.
  • [29] H. Triebel, Theory of Function Spaces, III, Monogr. Math. 100, Birkhäuser, Basel, 2006.
  • [30] H. Wang and Z. Liu, The wavelet characterization of Herz-type Hardy spaces with variable exponent, Ann. Funct. Anal. 3 (2012), no. 1, 128–141.
  • [31] S. Wu, A wavelet characterization for weighted Hardy spaces, Rev. Mat. Iberoam. 8 (1992), no. 3, 329–349.
  • [32] D. Yang, Y. Liang, and L. D. Ky, Real-Variable Theory of Musielak-Orlicz Hardy Spaces, Lecture Notes in Math. 2182, Springer, Cham, 2017.
  • [33] D. Yang and S. Yang, Local Hardy spaces of Musielak-Orlicz type and their applications, Sci. China Math. 55 (2012), no. 8, 1677–1720.
  • [34] D. Yang, C. Zhuo, and E. Nakai, Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. 29 (2016), no. 2, 245–270.