Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces

Abstract

Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and H_X({ℝ}^n) the associated Hardy-type space. In this article, we first establish the finite atomic characterization of H_X({ℝ}^n). As an application, we prove that the dual space of H_X({ℝ}^n) is the Campanato space associated with X. For any given \alpha \in (\mathrm{0},\mathrm{1}] and s\in {\mathbb{Z}}_{+}, using the atomic and the Littlewood–Paley function characterizations of H_X({ℝ}^n),we also establish its s-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area function S_{\alpha ,s},the intrinsic g-function g_{\alpha ,s},and the intrinsic g_{\lambda }^{\ast }-function g_{\lambda ,\alpha ,s}^{\ast }, where λ coincides with the best known range.

Keywords

Ball quasi-Banach function space / Hardy space / finite atomic characterization / Campanato space / intrinsic square function

References

[1] Bennett C, Sharpley R. Interpolation of Operators. Pure Appl Math, Vol 129. Boston: Academic Press, 1988

[2] Bownik M. Anisotropic Hardy Spaces and Wavelets. Mem Amer Math Soc, Vol 164, No 781. Providence: Amer Math Soc, 2003

[3] Chang D-C, Wang S, Yang D, Zhang Y. Littlewood–Paley characterizations of Hardytype spaces associated with ball quasi-Banach function spaces. Complex Anal Oper Theory, 2020, 14: Paper No 40 (33 pp)

[4] Cruz-Uribe D V, Fiorenza A. Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Appl Numer Harmon Anal. Heidelberg: Birkhäuser/Springer, 2013

[5] Cruz-Uribe D, Wang L-A D. Variable Hardy spaces. Indiana Univ Math J, 2014, 63: 447–493

[6] Fefferman C, Stein E M. H^p spaces of several variables. Acta Math, 1972, 129: 137–193

[7] Folland G B, Stein E M. Hardy Spaces on Homogeneous Groups. Math Notes, Vol 28. Princeton: Princeton Univ Press, 1982

[8] García-Cuerva J. Weighted H^p spaces. Dissertationes Math (Rozprawy Mat), 1979, 162: (63 pp)

[9] Grafakos L. Classical Fourier Analysis. 3rd ed. Grad Texts in Math, Vol 249. New York: Springer, 2014

[10] Huang J, Liu Y. Some characterizations of weighted Hardy spaces. J Math Anal Appl, 2010, 363: 121–127

[11] Huang L, Liu J, Yang D, Yuan W. Dual spaces of anisotropic mixed-norm Hardy spaces. Proc Amer Math Soc, 2019, 147: 1201–1215

[12] Jiao Y, Zuo Y, Zhou D, Wu L. Variable Hardy–Lorentz spaces Hp(⋅),q(ℝn).Math Nachr, 2019, 292: 309–349

[13] Lerner A K. Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals. Adv Math, 2011, 226: 3912–3926

[14] Lerner A K. On sharp aperture-weighted estimates for square functions. J Fourier Anal Appl, 2014, 20: 784–800

[15] Liang Y, Yang D. Musielak–Orlicz Campanato spaces and applications. J Math Anal Appl, 2013, 406: 307–322

[16] Liang Y, Yang D. Intrinsic square function characterizations of Musielak–Orlicz Hardy spaces. Trans Amer Math Soc, 2015, 367: 3225–3256

[17] Lindenstrauss J, Tzafriri L. Classical Banach Spaces. II. Function Spaces. Ergeb Math Grenzgeb, Vol 97. Berlin: Springer-Verlag, 1979

[18] Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal, 2012, 262: 3665–3748

[19] Nakai E, Sawano Y. Orlicz–Hardy spaces and their duals. Sci China Math, 2014, 57: 903–962

[20] Okada S, Ricker W J, Sánchez Pérez E A. Optimal Domain and Integral Extension of Operators. Acting in Function Spaces. Oper Theory Adv Appl, Vol 180. Basel: Birkhäuser, 2008

[21] Sawano Y. Theory of Besov Spaces. Dev Math, Vol 56. Singapore: Springer, 2018

[22] Sawano Y, Ho K-P, Yang D, Yang S. Hardy spaces for ball quasi-Banach function spaces. Dissertationes Math (Rozprawy Mat.), 2017, 525: 1–102

[23] Sawano Y, Tanaka H. The Fatou property of block spaces. J Math Sci Univ Tokyo, 2015, 22: 663–683

[24] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993

[25] Taibleson M H, Weiss G. The molecular characterization of certain Hardy spaces. Astérisque, 1980, 77: 67–149

[26] Wang F, Yang D, Yang S. Applications of Hardy spaces associated with ball quasi- Banach function spaces. Results Math, 2020, 75: Art 26 (58 pp)

[27] Wang H, Liu H. The intrinsic square function characterizations of weighted Hardy spaces. Illinois J Math, 2012, 56: 367–381

[28] Wang S, Yang D, Yuan W, Zhang Y. Weak Hardy-type spaces associated with ball quasi-Banach function spaces II: Littlewood–Paley characterizations and real interpolation. J Geom Anal, 2019,

[29] Wilson M. The intrinsic square function. Rev Mat Iberoam, 2007, 23: 771–791

[30] Wilson M. Weighted Littlewood–Paley Theory and Exponential-Square Integrability. Lecture Notes in Math, Vol 1924. Berlin: Springer, 2008

[31] Wilson M. How fast and in what sense(s) does the Calderón reproducing formula converge? J Fourier Anal Appl, 2010, 16: 768–785

[32] Wilson M. Convergence and stability of the Calderón reproducing formula in H¹ and BMO.J Fourier Anal Appl, 2011, 17: 801–820

[33] Yan X. Intrinsic square function characterizations of variable weak Hardy spaces. Taiwanese J Math, 2020, 24: 43–62

[34] Yan X. Intrinsic square function characterizations of weak Musielak–Orlicz Hardy spaces. Banach J Math Anal, 2019, 13: 969–988

[35] Yang D, Liang Y, Ky L D. Real-Variable Theory of Musielak–Orlicz Hardy Spaces. Lecture Notes in Math, Vol 2182. Cham: Springer-Verlag, 2017

[36] Zhang Y, Wang S, Yang D, Yuan W. Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: decompositions with applications to boundedness of Calderón–Zygmund operators. Sci China Math, 2020,

[37] Zhang Y, Yang D, Yuan W, Wang S. Real-variable characterizations of Orlicz-slice Hardy spaces. Anal Appl (Singap), 2019, 17: 597–664

[38] Zhuo C, Yang D, Liang Y. Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull Malays Math Sci Soc, 2016, 39: 1541–1577

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