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, Volume 28, Issue 1–3, pp 51–69 | Cite as

On pseudo-differential operators and smoothness of special Lie-group representations

  • H. O. Cordes


Two algebras of global pseudo-differential operators over ℝn are investigated, with corresponding classes of symbols A0=CB (all (x, ξ)-derivatives bounded over ℝ2n), and A1 (all finite applications of ∂xj, ∂ξj, and εpqpξqp∂xp on the symbol are in A0). The class A1 consists of classical symbols, i.e., ∂ α x β ξ a= 0((1+|ξ|)−|α|) for x ∈ Kc ℝ;n, K, compact. It is shown that a bounded operator A of 210C=L2(Rn) is a pseudo-differential operator with symbol a∈Aj if and only if the map A→G−1AG, G∈ gj is infinitely differentiable, from a certain Lie-group gj c GL(210C) to ℒ(210C) with operator norm. g0 is the Weyl (or Heisenberg) group. Extensions to operators of arbitrary order are discussed. Applications to follow in a subsequent paper.


Operator Norm Number Theory Algebraic Geometry Topological Group Subsequent Paper 
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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • H. O. Cordes
    • 1
  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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