Tohoku Mathematical Journal

Atomic decompositions of weighted Hardy spaces with variable exponents

Kwok-Pun Ho

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Abstract

We establish the atomic decompositions for the weighted Hardy spaces with variable exponents. These atomic decompositions also reveal some intrinsic structures of atomic decomposition for Hardy type spaces.

Article information

Source
Tohoku Math. J. (2), Volume 69, Number 3 (2017), 383-413.

Dates
First available in Project Euclid: 12 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1505181623

Digital Object Identifier
doi:10.2748/tmj/1505181623

Mathematical Reviews number (MathSciNet)
MR3695991

Zentralblatt MATH identifier
1378.42012

Subjects
Primary: 42B30: $H^p$-spaces
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 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
atomic decomposition weight Hardy spaces Littlewood-Paley theory maximal functions variable exponent analysis vector-valued maximal inequalities

Citation

Ho, Kwok-Pun. Atomic decompositions of weighted Hardy spaces with variable exponents. Tohoku Math. J. (2) 69 (2017), no. 3, 383--413. doi:10.2748/tmj/1505181623. https://projecteuclid.org/euclid.tmj/1505181623


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References

  • W. Abu-Shammala and A. Torchinsky, The Hardy-Lorentz spaces $H^{p,q}({\Bbb R}^n)$, Studia Math. 182 (2007), 283–294.
  • A. Almeida, J. Hasanov and S. Samko, Maximal and potential operators in variable exponent Morrey spaces, Georgian Math. J. 15 (2008), 195–208.
  • K. Andersen and R. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980), 19–31.
  • C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
  • H. Q. Bui, Weighted Hardy spaces, Math. Nachr. 103 (1981), 45–62.
  • D. SFO 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), 239–264.
  • D. Cruz-Uribe, J. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advance and Applications, Volume 215, Birkhäuser Basel, (2011).
  • D. Cruz-Uribe, A. Fiorenza and C. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), 223–238.
  • D. Cruz-Uribe, A. Fiorenza and C. Neugebauer, Weighted norm inequalities for the maximal operator on variable Lebesgue spaces, J. Math. Anal. Appl. 394 (2012), 744–760.
  • D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Birkhäuser, 2013.
  • L. Diening, Maximal functions on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004), 245–253.
  • L. Diening, Maximal function on Orlicz-Musielak spaces and generalized Lebesgue space, Bull. Sci. Math. 129 (2005), 657–700.
  • L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), 1731–1768.
  • L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta and T. Shimomura, Maximal functions on variable exponent spaces: limiting cases of the exponent, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 503–522.
  • L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev spaces with Variable Exponents, Springer, 2011.
  • M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777–799.
  • M. Frazier and B. Jawerth, The $\varphi$-transform and applications to distribution spaces, Function spaces and applications (Lund, 1986), 223–246, Lecture Notes in Math. 1302, Springer, Berlin, 1988.
  • M. Frazier and B. Jawerth, A Discrete transform and decomposition of distribution spaces, J. of Funct. Anal. 93, (1990), 34–170.
  • M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, BMS Regional Conf. Ser. in Math. 79. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991.
  • M. Frazier and B. Jawerth, Applications of the $\varphi$ and wavelet transforms to the theory of function spaces, Wavelets and their applications, 377–417, Jones and Bartlett, Boston, MA, 1992.
  • J. García-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116. Notas de Matemática [Mathematical Notes], 104. North-Holland Publishing Co., Amsterdam, 1985.
  • J. García-Cuerva, Weighted $H^p$ spaces, Dissertations Math. 162 (1979), 1–63.
  • L. Grafakos, Modern Fourier Analysis, second edition, Springer, 2009.
  • V. Guliyev, J. Hasanov and S. Samko, Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces, Math. Scand. 107 (2010), 285–304.
  • V. Guliyev and S. Samko, Maximal, potential, and singular operators in the genaralized variable exponent Morrey spaces on unbounded sets, J. Math Sci. 193 (2013), 228–248.
  • K.-P. Ho, Littlewood-Paley Spaces, Math. Scand. 108 (2011), 77–102.
  • K.-P. Ho, Wavelet bases in Littlewood-Paley spaces, East J. Approx. 17 (2011), 333–345.
  • 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), 375–406.
  • K.-P. Ho, Atomic decompositions of weighted Hardy-Morrey spaces, Hokkaido Math. J. 42 (2013), 131–157.
  • K.-P. Ho, Atomic decomposition of Hardy-Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), 31–62.
  • K.-P. Ho, Vector-valued operators with singular kernel and Triebel-Lizorkin-block spaces with variable exponents, Kyoto J. Math. 56 (2016), 97–124.
  • M. Izuki, E. Nakai and Y. Sawano, Hardy spaces with variable exponent, Harmonic analysis and nonlinear partial differential equations, 109–136, RIMS Kôkyûroku Bessatsu, B42, Res. Inst. Math. Sci. (RIMS), Kyoto, 2013.
  • H. Jia and H. Wang, Decomposition of Hardy-Morrey spaces, J. Math. Anal. Appl. 354 (2009), 99–110.
  • V. Kokilashvili and A. Meskhi, Boundedness of maximal and singular operators in Morrey spaces with variable exponent, Armen. J. Math. 1 (2008), 18–28.
  • D. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^p$ spaces, Trans. Amer. Math. Soc. 259 (1980), 235–254.
  • Y. Liang, Y. Sawano, T. Ullrich, D. Yang and W. Yuan, A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces, Dissertationes Math. 489 (2013), 114 pp.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II, Springer, 1996.
  • E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), 3665–3748.
  • E. Nakai and Y. Sawano, Orlicz-Hardy spaces and their duals, Sci. China Math. 57 (2014), 903–962.
  • H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd., Tokyo, 1950.
  • H. Nakano, Topology of Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, 1951.
  • A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}({\Bbb R}^n)$, Math. Inequal. Appl. 7 (2004), 255–265.
  • S. Okada, W. Ricker and E. Sánchez Pérez, Optimal Domain and Integral Extension of Operators, Oper. Theory Adv. Appl. 180. Birkhuser Verlag, Basel, 2008.
  • W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–211.
  • Y. Sawano and H. Tanaka, Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Math. Z. 257 (2007), 871–905.
  • Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures, Acta Math. Sin. (Engl. Ser.) 21 (2005), 1535–1544.
  • Y. Sawano, A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Acta Math. Sin. (Engl. Ser.) 25 (2009), 1223–1242.
  • Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), 123–148.
  • E. Stein and G. Weiss, On the thoery of harmonic functions of several variable, I: The theory of $H^p$ spaces, Acta Math. 103 (1960), 25–62.
  • E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.
  • J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Mathematics, 1381, Springer-Verlag, Berlin, 1989.
  • B. Viviani, An atomic decomposition of the predual of $BMO(\rho)$, Rev. Mat. Iberoamericana 3 (1987), 401–425.
  • W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Math. 2005, Springer-Verlag, Berlin, 2010.