Taiwanese Journal of Mathematics

Intrinsic Square Function Characterizations of Variable Weak Hardy Spaces

Xianjie Yan

Advance publication

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Abstract

Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, via using the atomic and Littlewood-Paley function characterizations of variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_{\lambda}^{\ast}$-function.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 20 pages.

Dates
First available in Project Euclid: 15 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1555315214

Digital Object Identifier
doi:10.11650/tjm/190401

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords
Hardy space variable exponent intrinsic square function atomic decomposition

Citation

Yan, Xianjie. Intrinsic Square Function Characterizations of Variable Weak Hardy Spaces. Taiwanese J. Math., advance publication, 15 April 2019. doi:10.11650/tjm/190401. https://projecteuclid.org/euclid.twjm/1555315214


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References

  • W. Abu-Shammala and A. Torchinsky, The Hardy-Lorentz spaces $H^{p,q}(\mathbb{R}^n)$, Studia Math. 182 (2007), no. 3, 283–294.
  • N. Aguilera and C. Segovia, Weighted norm inequalities relating the $g_{\lambda}^{\ast}$ and the area functions, Studia Math. 61 (1977), no. 3, 293–303.
  • A. Almeida and A. Caetano, Atomic and molecular decompositions in variable exponent $2$-microlocal spaces and applications, J. Funct. Anal. 270 (2016), no. 5, 1888–1921.
  • A. Almeida and P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), no. 5, 1628–1655.
  • J. Álvarez, $H^p$ and weak $H^p$ continuity of Calderón-Zygmund type operators, in: Fourier Analysis (Orono, ME, 1992), 17–34, Lecture Notes in Pure and Appl. Math. 157, Dekker, New York, 1994.
  • ––––, Continuity properties for linear commutators of Calderón-Zygmund operators, Collect. Math. 49 (1998), no. 1, 17–31.
  • Z. Birnbaum and W. Orlicz, Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen, Studia Math. 3 (1931), no. 1, 1–67.
  • D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013.
  • D. Cruz-Uribe, A. Fiorenza, J. M. 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.
  • D. Cruz-Uribe and L.-A. D. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), no. 2, 447–493.
  • L. Diening, P. Harjulehto, P. Hästö and M. R\ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011.
  • L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), no. 6, 1731–1768.
  • Y. Ding, S. Lu and S. Shao, Integral operators with variable kernels on weak Hardy spaces, J. Math. Anal. Appl. 317 (2006), no. 1, 127–135.
  • C. Fefferman, N. M. Rivière and Y. Sagher, Interpolation between $H^p$ spaces: the real method, Trans. Amer. Math. Soc. 191 (1974), 75–81.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193.
  • R. Fefferman and F. Soria, The space $\operatorname{Weak} H^{1}$, Studia Math. 85 (1986), no. 1, 1–16.
  • G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton University Press, Princeton, N.J., 1982.
  • J. Huang and Y. Liu, Some characterizations of weighted Hardy spaces, J. Math. Anal. Appl. 363 (2010), no. 1, 121–127.
  • M. Izuki, E. Nakai and Y. Sawano, Function spaces with variable exponents–-an introduction, Sci. Math. Jpn. 77 (2014), no. 2, 187–315.
  • A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226 (2011), no. 5, 3912–3926.
  • ––––, On sharp aperture-weighted estimates for square functions, J. Fourier Anal. Appl. 20 (2014), no. 4, 784–800.
  • Y. Liang and D. Yang, Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3225–3256.
  • H.-p. Liu, The weak $H^p$ spaces on homogenous groups, in: Harmonic Analysis (Tianjin, 1988), 113-118, Lecture Notes in Mathematics 1494, Springer, Berlin, 1991.
  • J. Liu, D. Yang and W. Yuan, Anisotropic variable Hardy-Lorentz spaces and their real interpolation, J. Math. Anal. Appl. 456 (2017), no. 1, 356–393.
  • S. Z. Lu, Four Lectures on Real $H^p$ Spaces, World Scientific, River Edge, NJ, 1995.
  • E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748.
  • W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), no. 1, 200–211.
  • T. Quek and D. Yang, Calderón-Zygmund-type operators on weighted weak Hardy spaces over $\mathbb{R}^n$, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 1, 141–160.
  • 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.
  • H. Wang, Boundedness of intrinsic square functions on the weighted weak Hardy spaces, Integral Equations Operator Theory 75 (2013), no. 1, 135–149.
  • ––––, Boundedness of several integral operators with bounded variable kernels on Hardy and weak Hardy spaces, Internat. J. Math. 24 (2013), no. 12, 1350095, 22 pp.
  • H. Wang and H. Liu, The intrinsic square function characterizations of weighted Hardy spaces, Illinois J. Math. 56 (2012), no. 2, 367–381.
  • M. Wilson, The intrinsic square function, Rev. Mat. Iberoam. 23 (2007), no. 3, 771–791.
  • ––––, Weighted Littlewood-Paley Theory and Exponential-square Integrability, Lecture Notes in Mathematics 1924, Springer, Berlin, 2008.
  • ––––, How fast and in what sense(s) does the Calderón reproducing formula converge? J. Fourier Anal. Appl. 16 (2010), no. 5, 768–785.
  • ––––, Convergence and stability of the Calderón reproducing formula in $H^1$ and $BMO$, J. Fourier Anal. Appl. 17 (2011), no. 5, 801–820.
  • L. Wu, D. Zhou, C. Zhuo and Y. Jiao, Riesz transform characterizations of variable Hardy-Lorentz spaces, Rev. Mat. Complut. 31 (2018), no. 3, 747–780.
  • J. Xu, Variable Besov and Triebel-Lizorkin spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 2, 511–522.
  • X. Yan, Intrinsic square function characterizations of weak Musielak-Orlicz Hardy spaces, Accepted.
  • X. Yan, D. Yang, W. Yuan and C. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal. 271 (2016), no. 10, 2822–2887.
  • D. Yang, W. Yuan and C. Zhuo, A survey on some variable function spaces, in: Function Spaces and Inequalities, 299–335, Springer Proc. Math. Stat. 206, Springer, Singapore, 2017.
  • 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.
  • D. Yang, C. Zhuo and W. Yuan, Triebel-Lizorkin type spaces with variable exponents, Banach J. Math. Anal. 9 (2015), no. 4, 146–202.
  • ––––, Besov-type spaces with variable smoothness and integrability, J. Funct. Anal. 269 (2015), no. 6, 1840–1898.
  • C. Zhuo, D. Yang and Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1541–1577.
  • C. Zhuo, D. Yang and W. Yuan, Interpolation between $H^{p(\cdot)}(\mathbb{R}^n)$ and $L^{\infty}(\mathbb{R}^n)$: real method, J. Geom. Anal. 28 (2018), no. 3, 2288–2311.