Integral Equations and Operator Theory

, Volume 26, Issue 1, pp 1–13 | Cite as

The convolution equation of Choquet and Deny on nilpotent groups

  • Cho-Ho Chu
  • Titus Hilberdink
Article

Abstract

G. Choquet and J. Deny have characterized the positive solutions μ of the convolution equation σ*μ=μ of measures on locally compact abelian groups, for a given positive measure σ. By elementary methods, we extend their characterization to locally compact nilpotent groups which complements the various existing results on the equation, and we work out the solutions μ explicitly for the Heisenberg groups and some nilpotent matrix groups, by finding all the exponential functions on these groups.

1991 Mathematics Subject Classification

Primary 43A05 46A55 45E10 45N05 Secondary 22E25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit, Wiener's tauberian theorem inL 1(G//K) and harmonic functions in the unit disk.Bull. Amer. Math. Soc. 32 (1995) 43–49.Google Scholar
  2. [2]
    G. Choquet, Lectures on Analysis, Vol.I,II, W.A. Benjamin, New York, 1969.Google Scholar
  3. [3]
    G. Choquet and J. Deny, Sur l'équation de convolution μ=μ*σ,C.R. Acad. Sc. Paris 250 (1960) 779–801.Google Scholar
  4. [4]
    C.-H. Chu and K.-S. Lau, Solutions of the operator-valued integrated Cauchy functional equation,J. Operator theory 32 (1994) 157–183.Google Scholar
  5. [5]
    P.L. Davies and D.N. Shanbhag, A generalization of a theorem of Deny with application in characterization theory,Quart. J. Oxford 38 (1987) 13–34.Google Scholar
  6. [6]
    J. Deny, Sur l'équation de convolution μ*σ=μ,Sémin. Théor. Potentiel de M. Brelot, Paris 1960.Google Scholar
  7. [7]
    J. D. Doob, J. Snell and R. E. Williamson, Application of boundary theory to sums of independent random variables,Contribution to Prob. and Stat., Stanford Univ. Press (1960) 182–197.Google Scholar
  8. [8]
    E. B. Dynkin and M. B. Malyutov, Random walks on groups with a finite number of generators,Soviet Math. Doklady 2 (1961) 399–402.Google Scholar
  9. [9]
    S. R. Foguel, On iterates of convolutions,Proc. Amer. Math. Soc. 47 (1978) 368–370.Google Scholar
  10. [10]
    H. H. Furstenberg, Poisson formula for semi-simple Lie groups.Ann. of Math. 77 (1963) 335–386.Google Scholar
  11. [11]
    E. E. Granirer, On some properties of the Banach algebrasA p (G) for locally compact groups,Proc. Amer. Math. Soc. 95 (1985) 375–381.Google Scholar
  12. [12]
    Y. Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques,Bull. Soc. Math. France 101 (1973) 333–379.Google Scholar
  13. [13]
    E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol.I, Second edition, Springer-Verlag, Berlin 1979.Google Scholar
  14. [14]
    A. M. Kagan, Yu. V. Linnik and C. R. Rao, Characterization problems in mathematical statistics, John Wiley, New York, 1973.Google Scholar
  15. [15]
    J.-P. Kahane, Lectures on mean periodic functions, Tata Inst. Bombay, 1959.Google Scholar
  16. [16]
    V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy,Ann. of prob. 11 (1983) 457–490.Google Scholar
  17. [17]
    J. L. Kelly, I. Namioka et al., Linear topological spaces, Van Nostrand, Princeton, 1963.Google Scholar
  18. [18]
    K.-S. Lau and C. R. Rao, Integrated Cauchy functional equation and characterizations of the exponential law,Sankhya A44 (1982) 72–90.Google Scholar
  19. [19]
    K.-S. Lau, J. Wang and C.-H. Chu, Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures,Studia Math. (to appear)Google Scholar
  20. [20]
    K.-S. Lau and W.-B. Zeng, The convolution equation of Choquet and Deny on semi-groups,Studia Math. 97 (1990) 115–135.Google Scholar
  21. [21]
    G. A. Margulis, Positive harmonic functions on nilpotent groups.Soviet Math. Doklady 166 (1966) 241–244.Google Scholar
  22. [22]
    R. R. Phelps, Lectures on Choquet's theorem, Van Nostrand, Princeton, 1966.Google Scholar
  23. [23]
    B. Ramachandran and K.-S. Lau, Functional equations in probability theory, Academic Press, 1991.Google Scholar
  24. [24]
    L. Schwartz, Théorie génerale des fonctions moyennes-périodiques,Ann. of Math, 48 (1947) 857–929.Google Scholar
  25. [25]
    T. Ramsey and Y. Weit, Ergodic and mixing properties of measures on locally compact abelian groups,Proc. Amer. Math. Soc. 92 (1984) 519–520.Google Scholar
  26. [26]
    V. S. Varadarajan, Lie groups, Lie algebras and their representations, Springer-Verlag, Berlin, 1984.Google Scholar

Copyright information

© Birkhäuser Verlag 1996

Authors and Affiliations

  • Cho-Ho Chu
    • 1
  • Titus Hilberdink
    • 1
  1. 1.Goldsmiths CollegeUniversity of LondonLondonEngland

Personalised recommendations