Traces on pseudodifferential operators and sums of commutators

Abstract

The aim of this paper is to show that various known characterizations of traces on classical pseudodifferential operators can actually be obtained by very elementary considerations on pseudodifferential operators, using only basic properties of these operators. Thereby, we give a unified treatment of the determinations of the space of traces (i) on ΨDOs of non-integer order or of regular parity-class, (ii) on integer order ΨDOs, (iii) on ΨDOs of non-positive orders in dimension ≥ 2, and (iv) on ΨDOs of non-positive orders in dimension 1.