Russian Journal of Mathematical Physics

, Volume 26, Issue 1, pp 9–31 | Cite as

Some Estimates for p-Adic Rough Multilinear Hausdorff Operators and Commutators on Weighted Morrey–Herz Type Spaces

  • N. M. ChuongEmail author
  • D. V. DuongEmail author
  • K. H. DungEmail author


The aim of this paper is to introduce and study the boundedness of a new class of p-adic rough multilinear Hausdorff operators on the product of Herz, central Morrey and Morrey–Herz spaces with both power weights and Muckenhoupt weights. We also establish the boundedness for the commutators of p-adic rough multilinear Hausdorff operators on the weighted spaces with symbols in central BMO space.


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  1. 1.
    S. Albeverio, A.Yu. Khrennikov, and V. M. Shelkovich, “Harmonic Analysis in the p–Adic Lizorkin Spaces: Fractional Operators, Pseudo–Differential Equations, p–Wavelets, Tauberian Theorems,” J. Fourier Anal. Appl. 12 (4), 393–425 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. V. Avetisov, A. H. Bikulov, S. V. Kozyrev, and V. A. Osipov, “p–Adic Models of Ultrametric Diffusion Constrained by Hierarchical Energy Landscapes,” J. Phys. A 35, 177–189 (2002).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. V. Avetisov, A. H. Bikulov, and V. A. Osipov, “p–Adic Description of Characteristic Relaxation in Complex Systems,” J. Phys. A 36, 4239–4246 (2003).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    K. F. Andersen and B. Muckenhoupt, “Weighted Weak Type Hardy Inequalities with Applications to Hilbert Transforms and Maximal Functions,” Studia Math. 72 (1), 9–26 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    O. V. Beloshapka, “Feynman Formulas for the Schrödinger Equations with the Vladimirov Operator,” Russ. J. Math. Phys. 17 (3), 267–271 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    C. Carton–Lebrun and M. Fosset, “Moyennes et quotients de Taylor dans BMO,” Bull. Soc. Roy. Sci. Liége 53 (2), 85–87 (1994).MathSciNetzbMATHGoogle Scholar
  7. 7.
    R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization Theorems for Hardy Spaces in Several Variables,” Ann. Math. 103, 611–635 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Chen, D. Fan and J. Li, “Hausdorff Operators on Function Spaces,” Chinese Ann. Math. Ser. B. 33, 537–556 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. M. Chuong, Pseudodifferential Operators and Wavelets over Real and pAdic Fields (Springer–Basel, 2018).CrossRefzbMATHGoogle Scholar
  10. 10.
    N. M. Chuong, Yu. V. Egorov, A. Yu. Khrennikov, Y. Meyer, and D. Mumford, Harmonic, Wavelet and pAdic Analysis (World Scientific, 2007).CrossRefzbMATHGoogle Scholar
  11. 11.
    N. M. Chuong, and D. V. Duong, “Wavelet Bases in the Lebesgue Spaces on the Field of p–Adic Numbers,” p−Adic numbers, Ultrametric Anal. Appl. 5 (2), 106–121 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    N. M. Chuong and D. V. Duong, “Weighted Hardy–Littlewood Operators and Commutators on p−Adic Functional Spaces,” p−Adic numbers, Ultrametric Anal. Appl. 5 (1), 65–82 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    N. M. Chuong and D. V. Duong, “The p–Adic Weighted Hardy–Cesàro Operators on Weighted Morrey–Herz Space,” p−Adic numbers, Ultrametric Anal. Appl. 8 (3), 204–216 (2016).CrossRefzbMATHGoogle Scholar
  14. 14.
    N. M. Chuong and N. V. Co, “The Cauchy Problem for a Class of Pseudo–Differential Equations over p–Adic Field,” J. Math. Anal. Appl. 340 (1), 629–643 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    N. M. Chuong and H. D. Hung, “Maximal Functions and Weighted Norm Inequalities on Local Fields,” Appl. Comput. Harmon. Anal. 29, 272–286 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, and I. V. Volovich, “On p−Adic Mathematical Physics,” p−Adic numbers, Ultrametric Anal. Appl. 1 (1), 1–17 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. Grafakos, Modern Fourier Analysis (Second Edition, Springer, 2008).zbMATHGoogle Scholar
  18. 18.
    H. D. Hung, “The p–Adic Weighted Hardy–Cesàro Operator and an Application to Discrete Hardy Inequalities,” J. Math. Anal. Appl. 409, 868–879 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Hytönen, C. Pérez, and E. Rela, “Sharp Reverse Hölder Property for A∞ Weights on Spaces of Homogeneous Type,” J. Funct. Anal. 263, 3883–3899 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    S. Indratno, D. Maldonado, and S. Silwal, “A Visual Formalism for Weights Satisfying Reverse Inequalities,” Expo. Math. 33, 1–29 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    S. Haran, “Riesz Potentials and Explicit Sums in Arithmetic,” Invent. Math. 101, 697–703 (1990).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. Haran, “Analytic Potential Theory over the p–Adics,” Ann. Inst. Fourier (Grenoble) 43 (4), 905–944 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A.Yu. Khrennikov, pAdic Valued Distributions in Mathematical Physics (Kluwer Academic Publishers, Dordrecht–Boston–London, 1994).CrossRefzbMATHGoogle Scholar
  24. 24.
    A.Yu. Khrennikov, V. M. Shelkovich, and M. Skopina, “p–Adic Refinable Functions and MRA–Based Wavelets,” J. Approx. Theory 161, 226–238 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    S. V. Kozyrev, “Methods and Applications of Ultrametric and p–Adic Analysis: From Wavelet Theory to Biophysics,” Proc. Steklov Inst. Math. 274, 1–84 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Z. W. Fu, Z. G. Liu, and S. Z. Lu, “Commutators of Weighted Hardy Operators,” Proc. Amer. Math. Soc. 137, 3319–3328 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Z. W. Fu, Q. Y. Wu, and S. Z. Lu, “Sharp Estimates of p–Adic Hardy and Hardy–Littlewood–P´olya Operators,” Acta Math. Sin. 29, 137–150 (2013).CrossRefzbMATHGoogle Scholar
  28. 28.
    S. Lu, Y. Ding, and D. Yan, Singular Integrals and Related Topics (World Scientific Publishing Company, Singapore, 2007).CrossRefzbMATHGoogle Scholar
  29. 29.
    C. Morrey, “On the Solutions of Quasi–Linear Elliptic Partial Differential Equations,” Trans. Amer. Math. Soc. 43, 126–166 (1938).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    B. Muckenhoupt, “Weighted Norm Inequalities for the Hardy Maximal Function,” Trans. Amer. Math. Soc. 165, 207–226 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    J. Ruan, D. Fan, and Q. Wu, “Weighted Herz Space Estimates for Hausdorff Operators on the Heisenberg Group,” Banach J. Math. Anal. 11, 513–535 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. Ruan, D. Fan, and Q. Wu, “Weighted Morrey Estimates for Hausdorf Op–erator and Its Commutator on the Heisenberg Group,” Math. Inequal. Appl. 22 (1), 303–329 (2019).Google Scholar
  33. 33.
    K. S. Rim and J. Lee, “Estimates ofWeighted Hardy–Littlewood Averages on the p–Adic Vector Space,” J. Math. Anal. Appl. 324 (2), 1470–1477 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Elias M. Stein, Harmonic Analysis, Real–Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, 1993).zbMATHGoogle Scholar
  35. 35.
    C. Tang, F. Xue, and Y. Zhou, “Commutators of Weighted Hardy Operators on Herz–Type Spaces,” Ann. Polon. Math. 101, 267–273 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    V. S. Varadarajan, “Path Integrals for a Class of p–Adic Schrödinger Equations,” Lett. Math. Phys. 39, 97–106 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    V. S. Vladimirov, “Tables of Integrals of Complex–Valued Functions of p–Adic Arguments,” Proc. Steklov Inst. Math. 284 (2), 1–59 (2014).MathSciNetCrossRefGoogle Scholar
  38. 38.
    V. S. Vladimirov and I. V. Volovich, “p–Adic Quantum Mechanics,” Comm. Math. Phys. 123, 659–676 (1989).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, pAdic Analysis and Mathematical Physis (World Scientific, 1994).CrossRefzbMATHGoogle Scholar
  40. 40.
    S. S. Volosivets, “Multidimensional Hausdorff Operator on p–Adic Field,” p–Adic numbers, Ultrametric Anal. Appl. 2, 252–259 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    S. S. Volosivets, “Hausdorff Operator of Special Kind in Morrey and Herz p–Adic Spaces,” p–Adic numbers, Ultrametric Anal. Appl. 4, 222–230 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    S. S. Volosivets, “Hausdorff Operators on p–Adic Linear Spaces and Their Properties in Hardy, BMO, and Hölder Spaces,” Math. Notes 3, 382–391 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    S. S. Volosivets, “Weak and Strong Estimates for Rough Hausdorff Type Operator Defined on p–Adic Linear Space,” p–Adic numbers, Ultrametric Anal. Appl. 9 (3), 236–241 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Q. Y. Wu, L. Mi, and Z. W. Fu, “Boundedness of p–Adic Hardy Operators and Their Commutators on p–Adic Central Morrey and BMO Spaces,” J. Funct. Spaces Appl. (2013), Article ID 359193, 10 pages.Google Scholar
  45. 45.
    J. Xiao, “L p and BMO Bounds of Weighted Hardy–Littlewood Averages,” J. Math. Anal. Appl. 262, 660–666 (2001).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam
  2. 2.School of MathematicsMientrung University of Civil EngineeringPhu YenVietnam
  3. 3.School of MathematicsUniversity of Transport and CommunicationsHanoiVietnam

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