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Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function

  • Sergio AlbeverioEmail author
  • Sergei V. Kozyrev
Research Articles

Abstract

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree T (O p ) of balls in O p . In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree T (O p ) and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.

p-adic pseudodifferential operators p-adic wavelets groups of automorphisms of trees 

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References

  1. 1.
    V. S. Vladimirov, I. V. Volovich, Ye. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).Google Scholar
  2. 2.
    A. N. Kochubei, Pseudo-Differential Equations and Stochastics over non-Archimedean Fields (Marcel Dekker, New York, Basel, 2001).zbMATHGoogle Scholar
  3. 3.
    S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izvestiya: Mathematics 66(2), 367–376 (2002). arXiv:math-ph/0012019zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. V. Kozyrev, “p-Adic pseudodifferential operators and p-adic wavelets,” Theor. Math. Phys. 138(3), 322–332 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Yu. Khrennikov and S. V. Kozyrev, “Pseudodifferential operators on ultrametric spaces and ultrametric wavelets,” Izvestiya: Mathematics 69(5), 989–1003 (2005); arXiv:math-ph/0412062.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets on ultrametric spaces,” Appl. Comp. Harm. Anal. 19, 61–76 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferenial operators,” Sbornik Mathematics 198(1), 103–126 (2007); arXiv:math-ph/0412082.CrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Albeverio and S. V. Kozyrev, “Multidimensional basis of p-adic wavelets and representation theory,” p-Adic Numbers, Ultrametric Analysis and Applications 1(3), 181–189 (2009); arXiv:0903.0461.CrossRefGoogle Scholar
  9. 9.
    B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adic mathematical physics,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 1–17 (2009); arXiv:0904.4205.CrossRefGoogle Scholar
  10. 10.
    A. Yu. Khrennikov, Information Dynamics in Cognitive, Psychological and Anomalous Phenomena, Series in Fundamental Theories of Physics (Kluwer, Dordrecht, 2004).Google Scholar
  11. 11.
    G. I. Olshansky, “Classification of irreducible representations of groups of automorphisms of Bruhat-Tits trees,” Func. Anal. Appl. 11(1), 26–34 (1997).CrossRefGoogle Scholar
  12. 12.
    Yu. A. Neretin, “On combinatorial analogs of the group of diffeomorphisms of the circle,” Izvestiya Mathematics 41(2), 337 (1993).MathSciNetGoogle Scholar
  13. 13.
    J.-P. Serre, Arbres, amalgames, SL(2), Ast érisque 46 (Societé mathématique de France, Paris, 1977).Google Scholar
  14. 14.
    J.-P. Serre, Trees (Springer, 1980 and 2003).Google Scholar
  15. 15.
    P. Cartier, “Harmonic analysis on trees,” in Harmonic Analysis on Homogeneous Spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), pp. 419–424 (Amer. Math. Soc., Providence, R.I., 1973).Google Scholar
  16. 16.
    P. Cartier, “Representations of p-adic groups: a survey,” in Automorphic Forms, Representations and L-Functions, (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 111–155, (Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979).Google Scholar
  17. 17.
    W. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis (Cambridge Univ. Press, 1984).Google Scholar

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© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Steklov Mathematical InstituteMoscowRussia

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