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

$$a \sim \sum\limits_{j = 0}^\infty {a_{m - j,} } a_{m - j,} (x,\xi ) = \sum\limits_{l = 0}^k {a_{m - j,l} (x,\xi )} \log ^l |\xi |,$$

where am-j,l is homogeneous in ξ of degree m-j. We call these symbols log-polyhomogeneous. We will explain why this algebra of pseudodifferential operators is natural.

We study log-polyhomogeneous functions on symplectic cones and generalize the symplectic residue of Guillemin to these functions. Similarly, as for homogeneous functions, for a log-polyhomogeneous function, this symplectic residue is an obstruction against being a sum of Poisson brackets.

For a pseudodifferential operator with log-polyhomogeneous symbol, A, and a classical elliptic pseudodifferential operator, P, we show that the generalized ζ-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 an analogue of the Kontsevich–Vishik trace also exists for our algebra. Our method also provides an alternative approach to the Kontsevich–Vishik trace.