Positivity

, Volume 11, Issue 1, pp 15–39

Gaussian Estimates of Order α and Lp-Spectral Independence of Generators of C0-Semigroups

Article

Abstract

We prove Lp-spectral independence for generators of C0-semigroups estimated by the positive C0-semigroup Open image in new window. In the preliminary process of the proof, we obtain the asymptotic expansion formula for the integral kernel of the C0-semigroup Open image in new window.

Mathematics Subject Classifications

47A25 47D03 47B65 41A60 

Keywords

Gaussian estimate Lp -spectrum positive semigroup integral kernel resolvent 

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References

  1. 1.
    G.E. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia of Mathematics and its Applications (no. 71), Cambridge University Press, Cambridge, 1999.Google Scholar
  2. 2.
    W. Arendt, Gaussian estimates and interpolation of the spectrum in Lp, Diff. Int. Equations 7(5) (1994), 1153–1168.Google Scholar
  3. 3.
    W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics (no. 96), Birkhäuser Verlag, Basel, 2001.Google Scholar
  4. 4.
    A.V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419–437.Google Scholar
  5. 5.
    E.B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989.Google Scholar
  6. 6.
    R. Hempel, J. Voigt, The spectrum of a Schrödinger operator in Open image in new window is p-independent, Communications in Math. Phy. 104 (1986), 243–250.Google Scholar
  7. 7.
    P.C. Kunstmann, Heat kernel estimates and Lp spectral independence of elliptic operators, Bulletin of the London Mathematical Society 31 (1999), 345–353.Google Scholar
  8. 8.
    P.C. Kunstmann, Kernel estimates and Lp-spectral independence of differential and integral operators, Operator theoretical methods (Timişoara, 1998), 197–211, The Theta Foundation, Bucharest, 2000.Google Scholar
  9. 9.
    S. Miyajima, M. Ishikawa, Generalization of Gaussian estimates and interpolation of the spectrum in Lp, SUT J. Math. 31(no. 2)(1995), 161–176.Google Scholar
  10. 10.
    R. Nagel (ed.), One-parameter semigroups of positive operators, Lecture Notes in Mathematics (no. 1184), Springer-Verlag, Berlin, 1986.Google Scholar
  11. 11.
    F.W.J. Olver, Error bounds for stationary phase approximations, SIAM J. Math. Anal. 5(1) (1974), 19–29.Google Scholar
  12. 12.
    H.H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin, 1974.Google Scholar
  13. 13.
    B. Simon, Schrödinger semigroups, Bulletin of the American Mathematical Society. N.S. 7 (1982), 447–526.Google Scholar
  14. 14.
    H. Tanabe, Equations of evolution, Pitman, London, 1979.Google Scholar
  15. 15.
    K. Yosida, Functional analysis (sixth edition), Springer-Verlag, Berlin, 1995.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceTokyo University of ScienceTokyoJapan

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