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The p-adic weighted Hardy-Cesàro operators on weighted Morrey-Herz space

P-Adic Numbers, Ultrametric Analysis, and Applications volume 8, pages 204–216 (2016)Cite this article

Abstract

In this article we give necessary and sufficient conditions for the boundedness of the weighted Hardy-Cesà ro operators which is associated to the parameter curve γ(t, x) = γ(t)x defined by \({U_{\psi ,\gamma }}f\left( x \right) = \int {\left( {\gamma \left( t \right)x} \right)} \psi \left( t \right)dt\) on the weighted Morrey-Herz space over the p-adic field. Especially, the corresponding operator norms are established in each case. These results actually extend those of K. S. Rim and J. Lee [27] and of the authors [9]. Moreover, the sufficient conditions of boundedness of commutators of p-adic weighted Hardy-Cesàro operator with symbols in the Lipschitz space on the weighted Morrey-Herz space are also established.

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References

  1. S. Albeverio and W. Karwowski, “A randomwalk on p-adics: the generator and its spectrum,” Stoch. Process. Appl. 53, 1–22 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  2. 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).

    Article  MathSciNet  MATH  Google Scholar 

  3. 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:Math. Gen. 35, 177–189 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. V. Avetisov, A. H. Bikulov and V. A. Osipov, “p-Adic description of characteristic relaxation in complex systems,” J. Phys. A:Math. Gen. 36, 4239–4246 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  5. 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).

    MathSciNet  MATH  Google Scholar 

  6. C. Carton-Lebrun and M. Fosset, “Moyennes et quotients de Taylor dans BMO,” Bull. Soc. Roy. Sci. Lié ge 53 (2), 85–87 (1994).

    MathSciNet  MATH  Google Scholar 

  7. R. R. Coifman, R. Rochberg and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Ann.Math. 103, 611–635 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  8. N. M. Chuong, Yu. V. Egorov, A. Yu. Khrennikov, Y. Meyer and D. Mumford, eds., Harmonic, Wavelet and p-Adic Analysis (World Scientific, 2007).

    Google Scholar 

  9. 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).

    Article  MathSciNet  MATH  Google Scholar 

  10. 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).

    Article  MathSciNet  MATH  Google Scholar 

  11. 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).

    Article  MathSciNet  MATH  Google Scholar 

  12. H. D.Hung, “The p-adic weightedHardy-Cesà ro operator and an application to discreteHardy inequalities,” J. Math. Anal. Appl. 409, 868–879 (2014).

    Article  MathSciNet  Google Scholar 

  13. M. Christ and L. Grafakos, “Best constants for two non-convolution inequalities,” Proc. Amer. Math. Soc. 123, 1687–1693 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  14. D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings (Springer-Verlag, Berlin, 2004).

    Book  MATH  Google Scholar 

  15. W. Faris, “Weak Lebesgue spaces and quantum mechanical binding,” Duke Math. J. 43, 365–373 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Haran, “Riesz potentials and explicit sums in arithmetic,” Invent Math. 101, 697–703 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Haran, “Analytic potential theory over the p-adics,” Ann. Inst. Fourier (Grenoble) 43 (4), 905–944 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  18. G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, second ed. (Cambridge Univ. Press, London, 1952).

    MATH  Google Scholar 

  19. A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer Acad. Publishers, Dordrecht-Boston-London, 1994).

    Book  MATH  Google Scholar 

  20. A. Yu. Khrennikov, V. M. Shelkovich and M. Skopina, “p-Adic refinable functions and MRA-based wavelets,” J. Approx. Theory 161, 226–238 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  21. 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).

    Article  MathSciNet  MATH  Google Scholar 

  22. Z. W. Fu, Z. G. Liu and S. Z. Lu, “Commutators of weighted Hardy operators,” Proc. Amer. Math. Soc. 137, 3319–3328 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. T. Lacey, “Commutators with Riesz potentials in one and several parameters,” Hokkaido Math. J. 36, 175–191 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Morrey, “On the solutions of quasi-linear elliptic partial differential equations,” Trans. Amer. Math. Soc. 43, 126–166 (1938).

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Paluszynski, “Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg andWeiss,” Indiana Univ. Math. J. 44, 1–17 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Perez, “Endpoints for commutators of singular integral operators,” J. Funct. Anal. 128, 163–185 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  27. K. S. Rim and J. Lee, “Estimates of weighted Hardy-Littlewood averages on the p-adic vector space,” J. Math. Anal. Appl. 324 (2), 1470–1477 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  28. C. Tang, F. Xue and Y. Zhou, “Commutators of weighted Hardy operators on Herz-type spaces,” Annales Polonici Mathematici 101.3, 267–273 (2011).

  29. M. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, 1975).

    MATH  Google Scholar 

  30. V. S. Varadarajan, “Path integrals for a class of p-adic Schro¨ dinger equations,” Lett.Math. Phys. 39, 97–106 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  31. V. S. Vladimirov and I. V. Volovich, “p-Adic quantum mechanics,” Comm. Math. Phys. 123, 659–676 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  32. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physis (World Scientific, 1994).

    Book  MATH  Google Scholar 

  33. S. S. Volosivets, “Multidimensional Hausdorff operator on p-adic field,” p-Adic Numbers Ultrametric Anal. Appl. 2, 252–259 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  34. S. S. Volosivets, “Hausdorff operator of special kind on p-adic field and BMO-type spaces,” p-Adic Numbers Ultrametric Anal. Appl. 3, 149–156 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  35. 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).

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Xiao, “Lp and BMO bounds of weighted Hardy-Littlewood averages,” J. Math. Anal. Appl. 262, 660–666 (2001).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam

    N. M. Chuong

  2. Department of Basic Sciences, Mientrung University of Civil Engineering, 24 Nguyen Du, Tuy Hoa City, Phu Yen, Vietnam

    D. V. Duong

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  1. N. M. Chuong
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  2. D. V. Duong
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Correspondence to N. M. Chuong.

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Chuong, N.M., Duong, D.V. The p-adic weighted Hardy-Cesàro operators on weighted Morrey-Herz space. P-Adic Num Ultrametr Anal Appl 8, 204–216 (2016). https://doi.org/10.1134/S207004661603002X

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  • DOI: https://doi.org/10.1134/S207004661603002X

Keywords

  • Weighted Hardy-Cesàro operator
  • commutator
  • weighted Morrey-Herz space
  • p-adic analysis