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On Some Properties of the Heisenberg Laplacian

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DOI: 10.4236/apm.2012.25051    3,527 Downloads   6,122 Views   Citations
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Let IHn be the (2n+1) -dimensional Heisenberg group and let Lα and be the sublaplacian and central element of the Lie algebra of IHn respectively. Forα=0 denote by L0=L the Heisenberg Laplacian and let K ∈Aut(IHn) be a compact subgroup of Au-tomorphism of IHn. In this paper, we give some properties of the Heisenberg Laplacian and prove that L and T generate the K-invariant universal enveloping algebra, U(hn)k of IHn.

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M. Egwe, "On Some Properties of the Heisenberg Laplacian," Advances in Pure Mathematics, Vol. 2 No. 5, 2012, pp. 354-357. doi: 10.4236/apm.2012.25051.


[1] G. B. Folland and E. M. Stein, “Estimate for the Complex and Analysis on the Heisenberg Group,” Communications on Pure and Applied Mathematics, Vol. 27, No. 4, 1974, pp. 429-522. doi:10.1002/cpa.3160270403
[2] R. Howe, “On the Role of the Heisenberg Group in Harmonic Analysis,” Bulletin of the American Mathematical Society, Vol. 3, No. 2, 1980, pp. 821-843.doi:10.1090/S0273-0979-1980-14825-9
[3] E. M. Stein, “Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals,” Princeton University Press, Princeton, 1993.
[4] G. B. Folland, “A Fundamental Solution for a Subelliptic Operator,” Bulletin of the American Mathematical Society, Vol. 79, No. 2, 1973, pp. 373-376. doi:10.1090/S0002-9904-1973-13171-4
[5] L. P. Rothschild, “Local Solvability of Left-Invariant Differential Operators on the Heisenberg Group,” Proceedings of the American Mathematical Society, Vol. 74, No. 2, 1979, pp. 383-388. doi:10.1090/S0002-9939-1979-0524323-X
[6] M. E. Egwe, “Aspects of Harmonic Analysis on the Heisenberg Group,” Ph.D. Thesis, University of Ibadan, Ibadan, 2010.
[7] H. Lewy, “An Example of a Smooth Linear Partial Differential Operator without Solution,” Annals of Mathematics, Vol. 66, No. 2, 1957, pp. 155-158. doi:10.2307/1970121
[8] U. N. Bassey and M. E. Egwe, “Non-Solvability of Heisenberg Laplacian by Factorization,” Journal of Mathematical Sciences, Vol. 21, No. 1, 2010, pp. 11-15.
[9] P. J. Olver, “Application of Lie Groups to Differential Equations,” Graduate Texts in Mathematics, Springer- Verlag, Berlin, 1986.
[10] S. Helgason, “Groups and Geometric Analysis: Integral Geometry, Differential Operators and Spherical Functions,” Academic Press Inc., New York, 1984.
[11] V. S. Varadarajan, “Lie Groups, Lie Algebras and Their Representations,” Springer-Verlag, Berlin, 1984.
[12] R. Strichartz, “Harmonic Analysis and Radon Transforms on the Heisenberg Group,” Journal of Functional Analysis, Vol. 96, No. 2, 1991, pp. 350-406.doi:10.1016/0022-1236(91)90066-E

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