Laguerre calculus and Paneitz operator on the Heisenberg group

Article
First Online:
Received:
Accepted:

Abstract

Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows:
$$ {\mathcal{P}_\alpha} = {\mathcal{L}_\alpha} \bar {\mathcal{L}_\alpha} = \frac{1} {4}\left[ {\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} } \right]^2 + \alpha ^2 T^2 $$
Here “Z j ” j=1 n is an orthonormal basis for the subbundle T (1,0) of the complex tangent bundle T ℂ(H n ) and T is the “missing direction”. The operator \( \mathcal{L}_\alpha \) is the sub-Laplacian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Λ α . We also construct projection operators and relative fundamental solution for the operator \( \mathcal{L}_\alpha \) while α ∈ Λ α .

Keywords

Paneitz operator Heisenberg group Laguerre calculus fundamental solution heat kernel spectrum 

MSC(2000)

35H20 53C44 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Folland G B, Stein E M. Estimates for the \( \bar \partial _b \) complex and analysis on the Heisenberg group. Comm Pure Appl Math, 27: 429–522 (1974)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beals R, Greiner P C. Calculus on Heisenberg Manifolds. In: Ann Math Studies, Vol. 119. Princeton: Princeton University Press, 1988Google Scholar
  3. 3.
    Berenstein C, Chang D C, Tie J. Laguerre Calculus and its Application on the Heisenberg Group. In: AMS/IP Series in Advanced Mathematics, Vol. 22. Cambridge: International Press, 2001Google Scholar
  4. 4.
    Lee J M. Pseudo-Einstein structure on CR manifolds. Amer J Math, 110: 157–178 (1988)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chang S-C, Tie J, Wu C-T. Subgradient estimate and Liouville-type theorems for the CR heat equation on Heisenberg groups. Preprint, 2008Google Scholar
  6. 6.
    Graham C R, Lee J M. Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains. Duke Math J, 57: 697–720 (1988)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hirachi K. Scalar pseudo-hermitian invariants and the Szegö kernel on 3-dimensional CR manifolds. Lect Notes Pure Appl Math, 143: 67–76 (1992)MathSciNetGoogle Scholar
  8. 8.
    Beals R, Gaveau B, Greiner P C. Complex Hamiltonian mechanics and parametrics for subelliptic Laplacians, I, II, III. Bull Sci Math, 121: 1–36, 97–149, 195–259 (1997)MATHMathSciNetGoogle Scholar
  9. 9.
    Calin O, Chang D C, Greiner P C. Geometric Analysis on the Heisenberg Group and Its Generalizations. In: AMS/IP Series in Advanced Mathematics, Vol. 40. Cambridge: International Press, 2007Google Scholar
  10. 10.
    Greiner P C. On the Laguerre calculus of left-invariant convolution operators on the Heisenberg group. Seminaire Goulaouic-Meyer-Schwartz, XI: 1–39 (1980–81)Google Scholar
  11. 11.
    Beals R, Gaveau B, Greiner P, et al. The Laguerre calculus on the Heisenberg group, II. Bull Sci Math, 110: 255–288 (1986)MathSciNetGoogle Scholar
  12. 12.
    Geller D. Fourier analysis on the Heisenberg group. Proc Natl Acad Sci USA, 74: 1328–1331 (1977)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Peetre J. The Weyl transform and Laguerre polynomials. Le Matematiche, 27: 301–323 (1972)MathSciNetGoogle Scholar
  14. 14.
    Folland G B. Harmonic Analysis in Phase Space. In: Annals of Math Studies, Vol. 122. Princeton: Princeton University Press, 1989Google Scholar
  15. 15.
    Stein E M. Harmonic Analysis-Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton University Press, 1993MATHGoogle Scholar
  16. 16.
    Kanwal R P. Generalized Functions: Theory and Applications, 3rd ed. Boston-Basel-Berlin: Birkhäuser, 2004MATHGoogle Scholar
  17. 17.
    Antimirov Ya M, Kolyshkin A A, Vaillancourt R. Complex Variables. San Diego-London-Boston-New York-Sydney-Tokyo-Toronto: Academis Press, 1997Google Scholar
  18. 18.
    Greiner P C, Stein E M. On the solvability of some differential operators of type □b. Ann Sc Norm Super Pisa Cl Sci (5), 4: 106–165 (1978)MathSciNetGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of MathematicsGeorgetown UniversityWashington D CUSA
  2. 2.Department of MathematicsNational Taiwan UniversityTaipei, TaiwanChina
  3. 3.Department of MathematicsUniversity of GeorgiaAthensUSA

Personalised recommendations