## The index of Fourier integral operators on manifolds with conical singularities.(English. Russian original)Zbl 1004.58015

Izv. Math. 65, No. 2, 329-355 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, No. 2, 127-154 (2001).
An elliptic Fourier integral operator is Fredholm in appropriate Sobolev spaces, and its index is a homotopy invariant of the canonical transformation between the cotangent bundles of two smooth compact manifolds and the principal symbol. Weinstein posed the problem of computing the index in classical terms. The index of such an operator called a quantized canonical transformation is independent of the choice of the principal symbol if either the dimension of the manifolds greater than or equal to 3 or the Maslov index on the graph of the canonical transformation is trivial mod 4.
For quantized canonical transformations of the cotangent bundle of a manifold with conical singularities into itself, Epstein-Melrose gave the index of the auxiliary fourier integral operator.
In this paper, the authors find an index formula for elliptic Fourier integral operators when the configuration space has canonical singularities. The index formular involves the index of an auxiliary Fourier integral operator on a smooth compact manifold and an explicitly specified invariant of the conormal symbol of the original Fourier integral operator.

### MSC:

 58J40 Pseudodifferential and Fourier integral operators on manifolds 58J20 Index theory and related fixed-point theorems on manifolds 35S30 Fourier integral operators applied to PDEs 47G30 Pseudodifferential operators
Full Text: