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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒮</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathscr{S}_{1}(L^{2}(G))} , as well as in the setting of the Schatten ideals <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒮</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathscr{S}_{r}(L^{2}(G))} , for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>r</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {r&gt;0} . In particular, we deduce Schatten characterizations of elliptic pseudo-differential operators of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ρ</m:mi> <m:mo>,</m:mo> <m:mi>δ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\rho,\delta)} -type for the large range <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>δ</m:mi> <m:mo>&lt;</m:mo> <m:mi>ρ</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0\leq\delta&lt;\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>×</m:mo> <m:mover accent="true"> <m:mi>G</m:mi> <m:mo>^</m:mo> </m:mover> </m:mrow> </m:math> {G\times\widehat{G}} , with the momentum variables belonging to the unitary dual <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>G</m:mi> <m:mo>^</m:mo> </m:mover> </m:math> {\widehat{G}} of G . In terms of the parameters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ρ</m:mi> <m:mo>,</m:mo> <m:mi>δ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(\rho,\delta)} , on the torus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝕋</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{T}^{n}} , we demonstrate the sharpness of our results show