A Muckenhoupt’s weight problem and vector valued maximal inequalities over local fields

  • Nguyen Minh Chuong
  • Ha Duy Hung
Research Articles


In this paper, an important and interesting Muckenhoupt’s problem over local fields was firstly solved. Weighted weak and strong type norm inequalities for the Fefferman-Stein vector-valued maximal operator were firstly established over local fields, too. These results are very useful. They could be applied in various mathematical areas, for instance, in theories of functions, of partial differential equations, in harmonic analysis.

Key words

Fefferman-Stein vector maximal operator weighted norm inequalities maximal inequalities p-adic fields local fields 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kenneth F. Andersen and Russel T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math., 69 (1980) 19–31.MathSciNetMATHGoogle Scholar
  2. 2.
    A. Benedek, A. P. Calderón and R. Panzone, “Convolution operators on Banach space valued functions,” Proc. Nat. Acad. Sc. USA. 48, 356–365 (1962).CrossRefMATHGoogle Scholar
  3. 3.
    A. Benedek, R. Panzone, “The spaces L p with mixed norm,” Duke Math. Jour. 28 301–324(1961).Google Scholar
  4. 4.
    L. Carleson and P. W. Jones, “Weighted norm inequalities and a theorem of Koosis,” Mittag-Leffler Rep. 2, (1981).Google Scholar
  5. 5.
    N. M. Chuong, Yu. V. Egorov, A. Khrennikov, Y. Meyer and D. Mumford, Harmonic, Wavelet and p-Adic Analysis (World Scientific, 2007).Google Scholar
  6. 6.
    N. M. Chuong, P. G. Ciarlet, P. Lax, D. Mumford and D. H. Phong, Advances in Deterministic and Stochastic Analysis (World Scientific, 2007).Google Scholar
  7. 7.
    N. M. Chuong and N. V. Co, “The multidimensional p-adic Green function,” Proc. Amer. Math. Soc. 127(3), 685–694 (1999).CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    N. M. Chuong and N. V. Co, “The Cauchy problem for a class of pseudodifferential equations over p-adic field,” J. Math. Anal. Appl. 340(1), 629–643 (2008).CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    N.M. Chuong and B. K. Cuong, “Convergence estimates of Galerkin-wavelet solutions to a Cauchy problem for a class of pseudodifferential equations,” Proc. Amer. Math. Soc. 132(12), 3589–3597 (2004).CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    N.M. Chuong, “Parabolic pseudodifferential operators of variable order in S.L. Sobolev spaces with weighted norms,” Dokl. Akad. Nauk SSSR 262(4), 804–807 (1982).MathSciNetGoogle Scholar
  11. 11.
    N. M. Chuong, “Degenerate parabolic pseudodifferential operators of variable order in S.L. Sobolev spaces with weighted norms,” Dokl. Akad. Nauk SSSR 268(5), 1055–1058 (1983).MathSciNetGoogle Scholar
  12. 12.
    N.M. Chuong and H. D. Hung, “Maximal functions and weighted norm inequalities on local fields,” Appl. Comput. Harmon. Anal. 29 272–286 (2010).CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    J. Diestel and Jr J. J. Uhl, Vector Measures (Amer. Math. Soc., Providence, R. I., 1977).MATHGoogle Scholar
  14. 14.
    J. L. Rubio de Francia, “Boundedness of maximal functions and singular integrals in weighted L p spaces,” Proc. Amer. Math. Soc. 83, 673–679 (1981).CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    C. Fefferman and E. M. Stein, “Some maximal inequalities,” Amer. J.Math. 93(1), 107–115 (1971).CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    L. Grafakos, Classical Fourier Analysis, (Sec. Edit., Springer, 2008).MATHGoogle Scholar
  17. 17.
    L. Grafakos, L. Liu and D. Yang, “Vector-valued singular integrals and maximal functions on spaces of homogeneous type,” Math. Scandinavica 104(2), 296–310 (2009).MathSciNetMATHGoogle Scholar
  18. 18.
    A. E. Gatto and C. E. Gutiérrez, “On weighted norm inequalities for the maximal function,” Studia Math. 83, 59–62 (1983).Google Scholar
  19. 19.
    R. A. Hunt, B. Muckenhoupt and R. Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transform,” Trans. Amer.Math. Soc. 176, 227–251 (1973).CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems, and Biological Models (Kluwer Acad. Publ., Dordrecht, 1997).MATHGoogle Scholar
  21. 21.
    A. Yu. Khrennikov and S. V. Kozyrev, “Pseudodifferential operators on ultrametric space and ultrametric wavelets,” Izvestia RAN: Ser. Mat. 69, 133–148 (2005) [in Russian]; Izvestia Math. 69, 989–1003 (2005) [English transl.].MathSciNetGoogle Scholar
  22. 22.
    A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets on ultrametric spaces,” Appl. Comput. Harmon. Anal. 19, 61–67 (2005).CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    A. Yu. Khrennikov and V. M. Shelkovich, “Non-Haar p-adic wavelets and their application to pseudodifferential operators and equations,” Appl. Comput. Harm. Anal. 28, 1–23 (2010).CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, Theory of p-Adic Distributions: Linear and Nonlinear Models (Oxford Univ. Press, Oxford, 2010).MATHGoogle Scholar
  25. 25.
    A. N. Kochubei, Pseudodifferential Equations and Stochastics over non-Archimedean Fields (Marcel Dekker, Inc. New York-Basel, 2001).Google Scholar
  26. 26.
    S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izv. Ross. Akad. Nauk: Ser. Math. 66(2), 149–158 (2002).MathSciNetGoogle Scholar
  27. 27.
    B. Muckenhoupt, “Weighted norm inequalities for classical operators,” in Proc. of Symposia in Pure Math., XXXV,part 1, 68–83.Google Scholar
  28. 28.
    K. Phillips and M. Taibleson, “Singular integrals in several variables over a local field,” Pacif. J. Math. 30, 209–231 (1969).MathSciNetMATHGoogle Scholar
  29. 29.
    E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton Univ. Press, 1993).Google Scholar
  30. 30.
    M. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, 1975).Google Scholar
  31. 31.
    A. Torchinsky, Real Variable Methods in Harmonic Analysis (Acad. Press, 1986).Google Scholar
  32. 32.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, 1994).Google Scholar
  33. 33.
    Wo-Sang Young, “Weighted norm inequalities for the Hardy-Littlewood maximal function,” Proc. Amer. Math. Soc. 85(1), 24–26 (1982).MathSciNetMATHGoogle Scholar
  34. 34.
    A. Zygmund, Trigonometries Series, Vol. I–II, 2nd ed. (Cambridge Univ. Press, Cambridge, 1959).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Institute of mathematicsVietnamese Academy of Science and TechnologyHanoiVietnam
  2. 2.Hanoi National University of EducationHanoiVietnam

Personalised recommendations