Advertisement

Annals of Global Analysis and Geometry

, Volume 28, Issue 3, pp 271–283 | Cite as

Weyl Transforms, the Heat Kernel and Green Function of a Degenerate Elliptic Operator

  • M. W. WongEmail author
Article

Abstract

We give a formula for the heat kernel of a degenerate elliptic partial differential operator L on 2 related to the Heisenberg group. The formula is derived by means of pseudo-differential operators of the Weyl type, {i.e.}, Weyl transforms, and the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for L2(2). Using the heat kernel, we give a formula for the Green function of L. Applications to the global hypoellipticity of L in the sense of tempered distributions, the ultracontractivity and hypercontractivity of the strongly continuous one-parameter semigroup etL, t > 0, are given.

Keywords

Weyl transforms heat kernel Green function global hypoellipticity ultracontractivity hypercontractivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boggiatto, P., Buzano, E., and Rodino, L.: Global Hypoellipticity and Spectral Theory, Akademie Verlag, Berlin, 1996.Google Scholar
  2. 2.
    Davies, E. B.: Heat Kernels and Spectral Theory, Cambridge University Press, New York, 1989.Google Scholar
  3. 3.
    Folland, G. B.: Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ, 1989.Google Scholar
  4. 4.
    Grossmann, A., Loupias, G. and Stein, E. M.: An algebra of pseudodifferential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1968), 343–368.Google Scholar
  5. 5.
    Magnus, W., Oberhettinger, F. and Soni, R. P.: Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin, 1964.Google Scholar
  6. 6.
    Nelson, E.: A quartic interaction in two dimensions, In: Mathematical Theory of Elementary Particles, MIT, Cambridge, MA, 1966, pp. 69–73.Google Scholar
  7. 7.
    Popivanov, P. R.: A link between small divisors and smoothness of the solutions of a class of partial differential equations, Ann. Global Anal. Geom. 1 (1983), 77–92.Google Scholar
  8. 8.
    Reed, M. and Simon, B.: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.Google Scholar
  9. 9.
    Shubin, M. A.: Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.Google Scholar
  10. 10.
    Simon, B.: The P(φ)2 Euclidean Quantum Field Theory, Princeton University Press, Princeton, NJ, 1974.Google Scholar
  11. 11.
    Thangavelu, S.: Lectures on Hermite and Laguerre Expansions, Princeton University Press, Princeton, NJ, 1993.Google Scholar
  12. 12.
    Thangavelu, S.: Harmonic Analysis on the Heisenberg Group, Birkhäuser, Boston, 1998.Google Scholar
  13. 13.
    Thangavelu, S.: An Introduction to the Uncertainty Principle: Hardy's Theorem on Lie Groups, Birkhäuser, Boston, 2004.Google Scholar
  14. 14.
    Wong, M. W.: Weyl Transforms, Springer-Verlag, Berlin, 1998.Google Scholar
  15. 15.
    Wong, M. W.: The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations