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


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.


Weyl transforms heat kernel Green function global hypoellipticity ultracontractivity hypercontractivity 


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  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

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© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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