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Pseudo-differential Operators on Compact Lie Groups

  • Michael Ruzhansky
  • Ville Turunen
Part of the Pseudo-Differential Operators book series (PDO, volume 2)

Abstract

In this chapter we develop a global theory of pseudo-differential operators on general compact Lie groups. As usual, S1,0 m (ℝ n × ℝ n ) ⊂ C (ℝ n × ℝ n ) refers to the Euclidean space symbol class, defined by the symbol inequalities
$$ \left| {\partial _\xi ^\alpha \partial _x^\beta p(x,\xi )} \right| \leqslant C(1 + \left| \xi \right|)^{m - \left| \alpha \right|} , $$
(10.1)
for all multi-indices α, β ∈ ℕ0 n , ℕ0 = {0}∪ℕ where the constant C is independent of x ξ ∈ ℝ n but may depend on α, β, p, m. On a compact Lie group G we define the class Ψ m (G) to be the usual Hörmander class of pseudo-differential operators of order m. Thus, the operator A belongs to Ψ m (G) if in (all) local coordinates operator A is a pseudo-differential operator on ℝ n with some symbol p(x, ξ) satisfying estimates (10.1), see Definition 5.2.11. Of course, symbol p depends on the local coordinate systems.

Keywords

Sobolev Space Convolution Operator Continuous Linear Operator Irreducible Unitary Representation Symbolic Calculus 
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

© Birkhäuser Verlag AG 2010

Authors and Affiliations

  • Michael Ruzhansky
    • 1
  • Ville Turunen
    • 2
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Institute of MathematicsHelsinki University of TechnologyFinland

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