ISSN:
1432-1785
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 εpq=ξp∂ξq−p∂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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01647964
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