On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols
Abstract
Description
We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ 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.
LaTeX2e, 35 pages; v2 16 Sept 1997, section on Kontsevich-Vishik added, Final version, 10 July 1998, minor corrections, to appear in Ann. Glob. Anal. Geom
LaTeX2e, 35 pages; v2 16 Sept 1997, section on Kontsevich-Vishik added, Final version, 10 July 1998, minor corrections, to appear in Ann. Glob. Anal. Geom