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manuscripta mathematica

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

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

  • H. O. Cordes
Article

Abstract

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.

Keywords

Operator Norm Number Theory Algebraic Geometry Topological Group Subsequent Paper 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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