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The Laguerre Calculus on the Heisenberg Group

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Special Functions: Group Theoretical Aspects and Applications

Part of the book series: Mathematics and Its Applications ((MAIA,volume 18))

Abstract

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

$${R^{2n + 1}} = \left\{ {\left( {{x_0},x'} \right)} \right\} = \left\{ {\left( {{x_0},{x_1},...,{x_{2n}}} \right)} \right\},$$
(1.1)

and with the group law

$$xy = \left( {{x_0},x'} \right)\left( {{y_0},y'} \right)$$
$$= \left( {{x_0} + {y_0} + \frac{1}{2}\sum\limits_{j = l}^n {{a_j}\left[ {{x_j}{y_{j + n}} - {x_{j + n}}{y_j}} \right]} ,x' + y'} \right).$$
(1.2)

.

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References

  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. Beals, R.W. and Greiner, P.C., “Non-elliptic differential operators of type □b”, (in preparation).

    Google Scholar 

  3. Folland, G.B., “A fundamental solution for a subelliptip operator”, Bull. Amer. Math. Soc. 79. (1973), pp. 373–376.

    Google Scholar 

  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. Geller, D., “Fourier analysis on the Heisenberg group. I. Schwartz space”, J. Func. Analysis 36 (1980), pp. 205–254.

    Google Scholar 

  6. Geller, D., “Local solvability and homogeneous distributions on the Heisenberg group”, Comm. PDE, 5 (5) (1980) pp. 475–560.

    Article  MathSciNet  MATH  Google Scholar 

  7. operators on the Heisenberg group”, Seminaire Goulaouic-Meyer-Schwartz, 1980-81, Expose no. XI, pp. 1–39.

    Google Scholar 

  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. 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. 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. 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.

    MathSciNet  Google Scholar 

  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. 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. Mikhlin, S.G., “Multidimensional singular integrals and integral equations”, Pergamon Press, 1965.

    Google Scholar 

  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. Seeley, R.T., “Elliptic Singular Integral Equations”, Amer. Math. Soc. Proc. Symp. Pure Math. 10 (1967), pp. 308–315.

    Google Scholar 

  17. Szegö, G., “Orthogonal polynomials”, Amer. Math. Soc. Colloquium Publ., V. 23, Amer. Math. Soc., Providence, R.I., 1939.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Yale University, USA

    R. W. Beals, P. C. Greiner & J. Vauthier

  2. The University of Toronto, Canada

    R. W. Beals, P. C. Greiner & J. Vauthier

  3. Universitè de Paris VI, France

    R. W. Beals, P. C. Greiner & J. Vauthier

Authors
  1. R. W. Beals
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  2. P. C. Greiner
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  3. J. Vauthier
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Editor information

Editors and Affiliations

  1. Mathematics Department, University of Wisconsin, Madison, USA

    R. A. Askey

  2. Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

    T. H. Koornwinder

  3. Department of Mathematics, University of Siegen, Germany

    W. Schempp

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© 1984 D. Reidel Publishing Company

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Beals, R.W., Greiner, P.C., Vauthier, J. (1984). The Laguerre Calculus on the Heisenberg Group. In: Askey, R.A., Koornwinder, T.H., Schempp, W. (eds) Special Functions: Group Theoretical Aspects and Applications. Mathematics and Its Applications, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9787-1_5

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  • DOI: https://doi.org/10.1007/978-94-010-9787-1_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0319-6

  • Online ISBN: 978-94-010-9787-1

  • eBook Packages: Springer Book Archive

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