Pseudo-differential Operators on Compact Lie Groups

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


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
$$ \left| {\partial _\xi ^\alpha \partial _x^\beta p(x,\xi )} \right| \leqslant C(1 + \left| \xi \right|)^{m - \left| \alpha \right|} , $$
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.


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