Schatten–von Neumann properties for Hörmander classes on compact Lie groups

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Abstract
Let G be a compact Lie group of dimension n . In this work we characterise the membership of classical pseudo-differential operators on G in the trace class ideal 𝒮 1 ⁢ ( L 2 ⁢ ( G ) ) {\mathscr{S}_{1}(L^{2}(G))} , as well as in the setting of the Schatten ideals 𝒮 r ⁢ ( L 2 ⁢ ( G ) ) {\mathscr{S}_{r}(L^{2}(G))} , for all r > 0 {r>0} . In particular, we deduce Schatten characterizations of elliptic pseudo-differential operators of ( ρ , δ ) {(\rho,\delta)} -type for the large range 0 ≤ δ < ρ ≤ 1 {0\leq\delta<\rho\leq 1} . Additional necessary and sufficient conditions are given in terms of the matrix-valued symbols of the operators, which are global functions on the phase space G × G ^ {G\times\widehat{G}} , with the momentum variables belonging to the unitary dual G ^ {\widehat{G}} of G . In terms of the parameters ( ρ , δ ) {(\rho,\delta)} , on the torus 𝕋 n {\mathbb{T}^{n}} , we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class Ψ 0 , 0 - ϰ ⁢ ( 𝕋 n ) {\Psi^{-\varkappa}_{0,0}(\mathbb{T}^{n})} , ϰ > 0 {\varkappa>0} , belonging to all the Schatten ideals. Additional order criteria are given in the setting of classical pseudo-differential operators. We present also some open problems in this setting.
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  • Engineering and Physical Sciences Research Council (EP/R003025/2)

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