Pseudodifferential calculus on manifolds with singular points (Q2715834)

The authors develop a calculus of pseudodifferential operators on a manifold with one-point singularities. The main tool is a Fourier-Laplace transform associated with some fixed, but otherwise arbitrary, diffeomorphism \( \delta \) of the positive half-line to the real line, and the transform ``quantizes'' the derivative \( -i (1/ \delta '(t)) (d/dt)\) in much the same way in which the Mellin transform quantizes the derivative \( -i t(d/dt)\), by transforming it into multiplication with the co-variable. The main emphasis is put on the local algebra at the ``singular point'' and in particular the notion of ellipticity and of Fredholmness (in appropriate Sobolev-type spaces) at the singular point is studied. Applications are given to derive an analytic index formula for elliptic operators of order zero near a singular point.