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Hardy Spaces Associated to Critical Herz Spaces with Variable Exponent

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Abstract

García-Cuerva (J Lond Math Soc (2) 39:499–513, 1989) has introduced Herz spaces associated to \({A^p}\) and studied atomic decomposition and its duality, where the space \({A^p}\) is a special case of Herz space. In this paper, we extend the atomic decomposition and duality results to the variable exponent settings.

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References

  1. Almeida A., Drihem D.: Maximal, potential and singular type operators on Herz spaces with variable exponents. J. Math. Anal. Appl. 394, 781–795 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bandaliev, R.A.: On a weighted variable spaces \({L_{p(x),\omega} \rm for 0\le {\it p}({\it x})\le 1}\) and weighted Hardy inequality. arXiv:1212.1695v1

  3. Chen Y.Z., Lau K.S.: Some new classes of Hardy spaces. J. Funct. Anal. 84, 255–278 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cruz-Uribe D., Fiorenza A., Neugebauer C.J.: The maximal function on variable \({L^p}\) spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: Corrections to: The maximal function on variable \({L^p}\) spaces [Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003)]. Ann. Acad. Sci. Fenn. Math. 29, 247–249 (2004)

  6. Diening L.: Maximal function on generalized Lebesgue spaces \({L^{p(\cdot)}}\). Math. Inequal. Appl. 7, 245–253 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011)

  8. Drihem, D., Seghiri, F.: Notes on the Herz-type Hardy spaces of variable smoothness and integrability. Math. Inequal. Appl. (to appear)

  9. Fefferman C., Stein E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feichtinger H., Weisz F.: Herz spaces and summability of Fourier transforms. Math. Nachr. 281(3), 309–324 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. García-Cuerva J.: Hardy spaces and Beurling algebras. J. Lond. Math. Soc. (2) 39, 499–513 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. García-Cuerva J., Herrero M.-J.L.: A theory of Hardy spaces associated to the Herz spaces. Proc. Lond. Math. Soc. (3) 69, 605–628 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North–Holland Mathematics Studies, vol. 116. North-Holland, Amsterdam (1985)

  14. Izuki M.: Herz and amalgam spaces with variable exponent, the Haar wavelets and greediness of the wavelet system. East J. Approx. 15, 87–109 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Izuki M.: Vector-valued inequalities on Herz spaces and characterizations of Herz–Sobolev spaces with variable exponent. Glas. Mat. 45, 475–503 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Izuki M.: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 36, 33–50 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Izuki M.: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo (2) 59, 199–213 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Izuki M.: Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent. Rend. Circ. Mat. Palermo (2) 59, 461–472 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Izuki M.: Fractional integrals on Herz–Morrey spaces with variable exponent. Hiroshima Math. J. 40, 343–355 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Izuki M., Noi T.: Duality of Besov, Triebel–Lizorkin and Herz spaces with variable exponents. Rend. Circ. Mat. Palermo (2) 63, 221–245 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kováčik O., Rákosník J.: On spaces \({L^{p(x)} \rm and {\it W^{k,p(x)}}}\). Czechoslov. Math. J. 41, 592–618 (1991)

    MATH  Google Scholar 

  22. Lu S., Wu Q., Yang D.: Boundedness of commutators on Hardy type space. Sci. China Ser. A. 45, 984–997 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu S., Xu L.: Boundedness of commutators related to Marcinkiewicz integrals on Hardy type spaces. Anal. Theory Appl. 20, 215–230 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Matsuoka, K.: On some weighted Herz spaces and the Hardy–Littlewood maximal operator. In: Proceedings of the International Symposium on Banach and Function Spaces II, Kitakyusyu, Japan, pp. 375–384 (2006)

  25. Rafeiro H., Samko S.: Riesz potential operators in continual variable exponents Herz spaces. Math. Nachr. 288, 465–475 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Samko S.: Differentiation and integration of variable order and the spaces \({L^{p(x)}}\). Contemp. Math. 212, 203–219 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stein E.M., Weiss G.: On the theory of harmonic functions of several variables I. The theory of \({H^p}\) spaces. Acta Math. 103, 25–62 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stein E.M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. University Press, Princeton (1971)

    MATH  Google Scholar 

  29. Wang H., Liu Z.: The Herz-type Hardy spaces with variable exponent and their applications. Taiwan. J. Math. 16, 1363–1389 (2012)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Education, Okayama University, 3-1-1 Tsushima-naka, Okayama, 700-8530, Japan

    Mitsuo Izuki

  2. Department of Mathematics, Chuo University, 13-27 Kasuga 1-chome, Bunkyoku, Tokyo, 112-8551, Japan

    Takahiro Noi

Authors
  1. Mitsuo Izuki
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  2. Takahiro Noi
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Correspondence to Takahiro Noi.

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Izuki, M., Noi, T. Hardy Spaces Associated to Critical Herz Spaces with Variable Exponent. Mediterr. J. Math. 13, 2981–3013 (2016). https://doi.org/10.1007/s00009-015-0668-2

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  • DOI: https://doi.org/10.1007/s00009-015-0668-2

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