Schatten-von Neumann properties for H\"ormander classes on compact Lie groups

TL;DR

This paper characterizes when classical pseudo-differential operators on compact Lie groups belong to Schatten classes, providing necessary and sufficient conditions based on their symbols, and explores the sharpness of these results on tori.

Contribution

It offers a comprehensive Schatten class characterization for pseudo-differential operators on compact Lie groups, including elliptic and exotic classes, with symbol-based criteria and sharpness demonstrations.

Findings

Characterization of Schatten class membership for pseudo-differential operators on compact Lie groups.

Necessary and sufficient conditions in terms of matrix-valued symbols.

Existence of atypical operators in exotic classes that belong to all Schatten ideals.

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 $S_{1}(L^2(G)),$ as well as in the setting of the Schatten ideals $S_{r}(L^2(G)),$ for all $r>0.$ In particular, we deduce Schatten characterisations of elliptic pseudo-differential operators of $(\rho,\delta)$-type for the large range $0\leq \delta<\rho\leq 1.$ 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 $G\times \widehat{G},$ with the momentum variables belonging to the unitary dual $\widehat{G}$ of $G$. In terms of the parameters $(\rho,\delta),$ on the torus $\mathbb{T}^n,$ we demonstrate the sharpness of our results showing the existence of atypical operators in the exotic class $\Psi^{-\varkappa}_{0,0}(\mathbb{T}^n),$ $\varkappa>0,$ 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.