Bulletin (New Series) of the American Mathematical Society

On the role of the Heisenberg group in harmonic analysis

Roger Howe

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Bull. Amer. Math. Soc. (N.S.), Volume 3, Number 2 (1980), 821-843.

First available in Project Euclid: 4 July 2007

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Primary: 42–02 35–02 22–02


Howe, Roger. On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 2, 821--843. https://projecteuclid.org/euclid.bams/1183547543

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  • [B] R. Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1-42.
  • [CV] A. Calderón and R. Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971), 374-378.
  • [Cr] P. Cartier, Quantum mechanical commutation relations and theta functions, Proc. Sympos. Pure Math., vol. 9, Amer. Math., Soc., Providence, R. I., 1966, pp. 361-383.
  • [D] R. G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972.
  • [GLS] A. Grossman, G. Loupias and E. Stein, An algebra of pseudo-differential operators and quantum mechanics in phase space, Ann. Inst. Fourier (Grenoble) 18 (1968), 343-368.
  • [Hr] L. Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979), 359-443.
  • [Hw1] R. Howe, On some results of Strichartz and of Rallis and Schiffmann, J. Functional Anal. 32 (1979), 297-303.
  • [Hw2] R. Howe, Remarks on Huyghens' principle (preprint).
  • [Hw3] R. Howe, Quantum mechanics and partial differential equations, J. Functional Anal. (to appear).
  • [Kr] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk. 17 (1962), 57-110; Russian Math. Surveys 17 (1962), 53-104.
  • [KS] A. Knapp and E. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489-578.
  • [KN] J. Kohn and L. Nirenberg, An algebra of pseudodifferential operators, Comm. Pure Appl. Math. 18 (1965), 269-305.
  • [L1] S. Lang, SL2 (R), Addison-Wesley, Reading, Mass., 1975.
  • [L2] S. Lang, Real analysis, Addison-Wesley, Reading, Mass., 1969.
  • [Ma] G. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265-311.
  • [MW] C. Moore and J. Wolf, Square integrable representations of nilpotent groups, Trans. Amer. Math. Soc. 185 (1973), 445-462.
  • [Mm] D. Mumford, On the equations defining abelian varieties. I, II, III, Invent. Math. 2 and 3, (1966) and (1967).
  • [RS] S. Rallis and G. Schiffmann, Weil representation. I. Intertwining distributions and discrete spectrum, Mem. Amer. Math. Soc. No. 231 (1980).
  • [Sg] I. Segal, Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand. 13 (1963), 31-43.
  • [Sh] D. Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149-167.
  • [St] R. Strichartz, Fourier transforms and non-compact rotation groups, Indiana Univ. Math. J. 24 (1974), 499-526.
  • [T] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967.
  • [V] U. Venugopalkrishna, Fredholm operators associated with strongly pseudoconvex domains in Cn, J. Functional Anal. 9 (1972), 349-372.
  • [vN]. J. von Neumann, Die eindeutigkeit der Schröderschen Operatoren, Math. Ann. 104 (1931), 570-578.
  • [Wi] A. Weil, Sur certains groupes d'opérateurs unitaires, Acta. Math. 111 (1964), 145-211.
  • [Wy1] H. Weyl, The theory of groups and quantum mechanics, Methuen, London, 1931.
  • [Wy2] H. Weyl, The classical groups, Princeton Univ. Press, Princeton, N. J., 1939.