Pseudo-differential Operators on Compact Lie Groups

Abstract

AbstractIn this chapter we develop a global theory of pseudo-differential operators on general compact Lie groups. As usual, S1,0m (ℝn × ℝn) ⊂ C∞ (ℝn × ℝn) 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 ∈ ℕ0n, ℕ0 = {0}∪ℕ where the constant C is independent of x ξ ∈ ℝn but may depend on α, β, p, m. On a compact Lie group G we define the class Ψm (G) to be the usual Hörmander class of pseudo-differential operators of order m. Thus, the operator A belongs to Ψm (G) if in (all) local coordinates operator A is a pseudo-differential operator on ℝn 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.KeywordsSobolev SpaceConvolution OperatorContinuous Linear OperatorIrreducible Unitary RepresentationSymbolic CalculusThese 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.