The Laguerre Calculus on the Heisenberg Group

  • R. W. Beals
  • P. C. Greiner
  • J. Vauthier
Part of the Mathematics and Its Applications book series (MAIA, volume 18)


The purpose of this article is to derive a multiplicative symbolic calculus for left-invariant convolution operators on the Heisenberg group. We let Hn denote the n-th Heisenberg group with underlying manifold
$${R^{2n + 1}} = \left\{ {\left( {{x_0},x'} \right)} \right\} = \left\{ {\left( {{x_0},{x_1},...,{x_{2n}}} \right)} \right\},$$
and with the group law
$$xy = \left( {{x_0},x'} \right)\left( {{y_0},y'} \right)$$
$$= \left( {{x_0} + {y_0} + \frac{1}{2}\sum\limits_{j = l}^n {{a_j}\left[ {{x_j}{y_{j + n}} - {x_{j + n}}{y_j}} \right]} ,x' + y'} \right).$$


HEISENBERG Group Convolution Operator Infinite Matrix Zygmund Operator Twisted Convolution 
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  1. [1]
    Beals, R.W. and Greiner, P.C., “Pseudo-differential operators associated to hyperplane bundles”, Bull. Sem. Mat. Torino, pp. 7–40, 1983.Google Scholar
  2. [2]
    Beals, R.W. and Greiner, P.C., “Non-elliptic differential operators of type □b”, (in preparation).Google Scholar
  3. [3]
    Folland, G.B., “A fundamental solution for a subelliptip operator”, Bull. Amer. Math. Soc. 79. (1973), pp. 373–376.Google Scholar
  4. [4]
    Folland, G.B. and Stein, E.M., “Estimates for the 3b- complex and analysis on the Heisenberg group”, Comm. Pure Appl. Math. 27 (1974), pp. 429–522.Google Scholar
  5. [5]
    Geller, D., “Fourier analysis on the Heisenberg group. I. Schwartz space”, J. Func. Analysis 36 (1980), pp. 205–254.Google Scholar
  6. [6]
    Geller, D., “Local solvability and homogeneous distributions on the Heisenberg group”, Comm. PDE, 5 (5) (1980) pp. 475–560.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7] Greiner, P.C., “On the Laguerre calculus of left-invariant convolution (pseudo-differential)
    operators on the Heisenberg group”, Seminaire Goulaouic-Meyer-Schwartz, 1980-81, Expose no. XI, pp. 1–39.Google Scholar
  8. [8]
    Greiner, P.C., Kohn, J.J. and Stein, E.M., “Necessary and sufficient conditions for the solvability of the Lewy equation”, Proc. Nat. Acad, of Sciences, U.S.A., 72 (1975), pp. 3287–3289.Google Scholar
  9. [9]
    Greiner, P.C. and Stein, E.M., “Estimates for the 3- Neumann problem”, Math. Notes Series, no. 19, Princeton Univ. Press, Princeton, N.J. 1977.Google Scholar
  10. [10]
    Greiner, P.C., and Stein, E.M., “On the solvability of some differential operators of type □b”, Proc. of the Seminar on Several Complex Variables, Cortona, Italy, 1976–1977, pp. 106–165.Google Scholar
  11. [11]
    Koranyi, A. and Vagi, S., “Singular integrals in homogeneous spaces and some problems of classical analysis” Ann. Sauola Norm. Sup. Pisa 25. (1971), pp. 575–648.MathSciNetGoogle Scholar
  12. [12]
    Lewy, H., “An example of a smooth linear partial differential equation without solution”, Ann. of Math., 66 (1957), pp. 155–158.Google Scholar
  13. [13]
    Mauceri, G., “The Weyl transform and bounded operators on Lp(Rn)”, Report no.54 of the Math. Inst, of the Univ. of Genova, 1980.Google Scholar
  14. [14]
    Mikhlin, S.G., “Multidimensional singular integrals and integral equations”, Pergamon Press, 1965.Google Scholar
  15. [15]
    Nagel, A. and Stein, E.M., “Lectures on pseudo-differential operators”, Math. Notes Series, no. 24, Princeton Univ. Press, Princeton, N.J. 1979.Google Scholar
  16. [16]
    Seeley, R.T., “Elliptic Singular Integral Equations”, Amer. Math. Soc. Proc. Symp. Pure Math. 10 (1967), pp. 308–315.Google Scholar
  17. [17]
    Szegö, G., “Orthogonal polynomials”, Amer. Math. Soc. Colloquium Publ., V. 23, Amer. Math. Soc., Providence, R.I., 1939.Google Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • R. W. Beals
    • 1
    • 2
    • 3
  • P. C. Greiner
    • 1
    • 2
    • 3
  • J. Vauthier
    • 1
    • 2
    • 3
  1. 1.Yale UniversityUSA
  2. 2.The University of TorontoCanada
  3. 3.Universitè de Paris VIFrance

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