On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols

TL;DR

This paper investigates the noncommutative residue for a class of pseudodifferential operators with log-polyhomogeneous symbols, exploring their algebraic properties, trace functionals, and meromorphic zeta functions.

Contribution

It introduces a natural algebra of pseudodifferential operators with log-polyhomogeneous symbols and constructs higher noncommutative residues and trace functionals for this algebra.

Findings

The zeta function $ ext{Tr}(AP^{-s})$ extends meromorphically with possible higher order poles.

The algebra admits a bigrading by order and log-power, but no nontrivial traces exist on the whole algebra.

An analogue of the Kontsevich-Vishik trace is established for this class of operators.

Abstract

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,\xi)=\sum_{l=0}^k a_{m-j,l}(x,\xi) \log^l|\xi|,$ where $a_{m-j,l}$ is homogeneous in $\xi$ of degree $m-j$. We will explain why this algebra of pseudodifferential operators is natural. For a pseudodifferential operator in this class, $A$, and a classical elliptic pseudodifferential operator, $P$, we show that the generalized zeta-function $\Tr(AP^{-s})$ has a meromorphic continuation to the whole complex plane, however possibly with higher order poles. Our algebra of operators has a bigrading given by the order and the highest log-power occuring in the symbol expansion. We construct "higher" noncommutative residue functionals on the subspaces given by the log-grading. However, in contrast to the classical case we prove that the whole algebra does not admit any nontrivial traces. Finally we show that the analogue of the Kontsevich-Vishik trace also exists on our algebra. Our method also provides an alternative approach to the Kontsevich-Vishik trace.