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

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.

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

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