Elliptic Differential Operators on Manifolds with Conical Singularities
by
Elmar Schrohe
Institut für Mathematik, Universität Potsdam, Postfach 601553, D-14415 Potsdam, Germany

We consider a manifold B with a conical singularity. Blowing up near the tip we obtain a manifold with boundary, on which we study Fuchs type degenerate differential operators.

These operators naturally act on scales of weighted Mellin L^p-Sobolev spaces, 1 < p < \infty. For many purposes, however, it is more convenient to consider them as unbounded operators on L^p(B). Under a rather mild ellipticity condition, one can characterize the domains of all possible closed extensions as sums of the Mellin Sobolev spaces and finite-dimensional spaces of asymptotics.

Stronger assumptions make it possible to determine the structure of the resolvents at least of the minimal and the maximal extension. With the help of this information, we can construct a family of complex powers and show the boundedness of the purely imaginary powers.

Date received: March 26, 2001