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On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols. (English) Zbl 0920.58047
Summary: We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion \[ a\sim \sum_{j=0}^\infty a_{m-j}, \quad a_{m-j}(x,\xi)= \sum_{l=0}^k a_{m-j,l} (x,\xi) \log^l| \xi| , \] where \(a_{m-j,l}\) is homogeneous in \(\xi\) of degree \(m-j\). We 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 \(\zeta\)-function \(\text{Tr} (AP^{-s})\) 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.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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