Summary
Given a matrixS∋sp (n, ℝ), one finds a second-order, bi-invariant differential operator □ s on the one-dimensional extensionG s of the Heisenberg groupH n induced byS. We construct solutions to certain Cauchy problems for □ s and fundamental solutions for these operators. This analysis is connected with the discussion of the “oscillator semigroup” by R. Howe.
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References
[AM] Abraham, R., Marsden, J.: Foundations of mechanics. Benjamin, Reading, Mass. 1978
[B] Banesti, F.: Résolubilité globale d'opérateurs différentiels invariants sur certains groupes de Lie. J. Funct. Anal.77, 261–308 (1988)
[C] Cowling, M.: Pointwise behaviour of solutions to Schrödinger equations. (Lect. Notes Math., vol. 992, pp. 83–90) Berlin Heidelberg New York: Springer 1983
[Di] Dixmier, J.: Enveloping algebras. North-Holland Math. Library, Amsterdam, vol. 14, 1977
[DK] Dahlberg, B.E.J., Kenig, C.E.: A note on the almost everywhere behaviour of solutions to the Schrödinger equation. (Lect. Notes Math., vol.908, pp. 205–209) Berlin Heidelberg New York: Springer 1982
[F] Folland, G.B.: Harmonic analysis in phase space. Princeton University Press. Ann. Math. Stud. 1989
[GeSt] Geller, D., Stein, E.M.: Singular convolution operators on the Heisenberg group. Bull. Am. Math. Soc.6, 99–103 (1982)
[GrSt] Greiner, P.C., Stein, E.M.: Estimates for the σ-Neumann problem. Math. Notes, Princeton University Press 1977
[GR] Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topies. North-Holland Mathematics Studies, vol. 116, 1985
[H1] Howe, R.: Quantum mechanics and partial differential equations. J. Funct. Anal.38, 188–254 (1980)
[H2] Howe, R.: The oscillator semigroup. Proc. Symp. Pure Math.48, 61–132 (1988)
[Hö] Hörmander, L.: The analysis of linear partial differential operators II. Berlin Heidelberg New York: Springer 1985
[MR] Müller, D., Ricci, F.: On the Laplace-Beltrami operator on the oscillator group. J. Reine Angew. Math.390, 193–207 (1988)
[P] Penney, R.C.: Non-elliptic Laplace equations on nilpotent Lie groups. Ann. Math.119, 309–385 (1984)
[PhSt] Phong, D.H., Stein, E.M.: Hilbert integrals, singular integrals, and Radon transform I. Acta Math.157, 99–157 (1986)
[ReS] Reed, M., Simon, B.: Methods of modern mathematical physics I. New York: Academic Press 1972
[RSj] Ricci, F., Sjögren, P.: Two-parameter maximal functions in the Heisenberg group. Math. Z.199, 565–575 (1988)
[S] Shale, D.: Linear symmetries of free boson fields. T.A.M.S.103, 149–167 (1962)
[St] Stein, E.M.: On limits of sequences of operators. Ann. Math.74, 140–170 (1961)
[T] Taylor, M.E.: Noncommutative harmonic analysis. Math. surveys and monographs, Am. Math. Soc.22, 1985
[W] Weil, A.: Sur certain groupes d'operateurs unitaires. Acta Math.111, 143–211 (1964)
[Y] Yosida, K.: Functional Analysis. Berlin Heidelberg New York: Springer 1980
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Oblatum 6-VII-1989
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Müller, D., Ricci, F. Analysis of second order differential operators on Heisenberg groups. I. Invent Math 101, 545–582 (1990). https://doi.org/10.1007/BF01231515
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DOI: https://doi.org/10.1007/BF01231515