Glasnik Matematicki, Vol. 45, No.2 (2010), 475-503.

VECTOR-VALUED INEQUALITIES ON HERZ SPACES AND CHARACTERIZATIONS OF HERZ-SOBOLEV SPACES WITH VARIABLE EXPONENT

Mitsuo Izuki

Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10 Nishi 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan
e-mail: mitsuo@math.sci.hokudai.ac.jp

Abstract.   Our first aim in this paper is to prove the vector-valued inequalities for some sublinear operators on Herz spaces with variable exponent. As an application, we obtain some equivalent norms and wavelet characterization of Herz-Sobolev spaces with variable exponent.

2000 Mathematics Subject Classification.   42B20, 42B35, 42C40, 46B15.

Key words and phrases.   Herz-Sobolev space with variable exponent, wavelet, vector-valued inequality.

Full text (PDF) (free access)

DOI: 10.3336/gm.45.2.14

References:

  1. A. Baernstein II and E. T. Sawyer, Embedding and multiplier theorems for Hp(Rn), Mem. Amer. Math. Soc. 53 (1985), no. 318.
    MathSciNet    

  2. D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. Pérez, The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), 239-264.
    MathSciNet    

  3. D. Cruz-Uribe, A. Fiorenza and C. J. Neugebauer, The maximal function on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238, and 29 (2004), 247-249.
    MathSciNet     %i
    MathSciNet    

  4. L. Diening, Maximal functions on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), 657-700.
    MathSciNet     CrossRef

  5. E. Hernández and G. Weiss, A first course on wavelets, CRC Press, Boca Raton, 1996.
    MathSciNet    

  6. E. Hernández, G. Weiss and D. Yang, The φ-transform and wavelet characterizations of Herz-type spaces, Collect. Math. 47 (1996), 285-320.
    MathSciNet    

  7. E. Hernández and D. Yang, Interpolation of Herz spaces and applications, Math. Nachr. 205 (1999), 69-87.
    MathSciNet    

  8. C. S. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968/1969), 283-323.
    MathSciNet    

  9. M. Izuki, Wavelets and modular inequalities in variable Lp spaces, Georgian Math. J. 15 (2008), 281-293.
    MathSciNet    

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

  11. M. Izuki, Wavelet characterization of Herz-Sobolev spaces with variable exponent, submitted.

  12. M. Izuki and K. Tachizawa, Wavelet characterizations of weighted Herz spaces, Sci. Math. Jpn. 67 (2008), 353-363.
    MathSciNet    

  13. S. E. Kelly, M. A. Kon and L. A. Raphael, Local convergence for wavelet expansions, J. Funct. Anal. 126 (1994), 102-138.
    MathSciNet     CrossRef

  14. T. S. Kopaliani, Greediness of the wavelet system in Lp(t)(R) spaces, East J. Approx. 14 (2008), 59-67.
    MathSciNet    

  15. O. Kováčik and J. Rákosn'ik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(116) (1991), 592-618.
    MathSciNet    

  16. D. S. Kurtz, Littlewood-Paley and multiplier theorems on weighted Lp spaces, Trans. Amer. Math. Soc. 259 (1980), 235-254.
    MathSciNet     CrossRef

  17. X. Li and D. Yang, Boundedness of some sublinear operators on Herz spaces, Illinois J. Math. 40 (1996), 484-501.
    MathSciNet     CrossRef

  18. S. Lu, K. Yabuta and D. Yang, Boundedness of some sublinear operators in weighted Herz-type spaces, Kodai Math. J. 23 (2000), 391-410.
    MathSciNet     CrossRef

  19. S. Lu and D. Yang, The decomposition of weighted Herz spaces on Rn and its application, Sci. China Ser. A 38 (1995), 147-158.
    MathSciNet    

  20. S. Lu and D. Yang, Herz-type Sobolev and Bessel potential spaces and their applications, Sci. China Ser. A 40 (1997), 113-129.
    MathSciNet     CrossRef

  21. Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992.
    MathSciNet    

  22. E. Nakai, N. Tomita and K. Yabuta, Density of the set of all infinitely differentiable functions with compact support in weighted Sobolev spaces, Sci. Math. Jpn. 60 (2004), 121-127.
    MathSciNet    

  23. A. Nekvinda, Hardy-Littlewood maximal operator on Lp(x)(Rn), Math. Inequal. Appl. 7 (2004), 255-265.
    MathSciNet    

  24. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.
    MathSciNet    

  25. L. Tang and D. Yang, Boundedness of vector-valued operators on weighted Herz spaces, Approx. Theory Appl. (N.S.) 16 (2000), 58-70.
    MathSciNet    

  26. H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
    MathSciNet    

  27. J. Xu and D. Yang, Applications of Herz-type Triebel-Lizorkin spaces, Acta Math. Sci. Ser. B Engl. Ed. 23 (2003), 328-338.
    MathSciNet    

  28. J. Xu and D. Yang, Herz-type Triebel-Lizorkin spaces. I, Acta Math. Sci. (Engl. Ser.) 21 (2005), 643-654.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page