Ellipticity and Invertibility in the Cone Algebra on Lp-Sobolev Spaces

Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of Lp-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces; it turns out to be independent of the choice of p. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators on Lp(B). Let B be a manifold with conical singularities. By deenition, B is a smooth (n + 1)-dimensional manifold outside a nite set of exceptional points. In a neighborhood of each point b in this collection, B has the structure of a cone whose cross-section, X b , is a smooth compact manifold of dimension n. Following the standard procedure, we blow up at each b. We obtain locally the cylinder 0; 1) X b and globally a manifold B with boundary which makes the analysis much more convenient. For simplicity, we assume that we only have one singularity. Fixing a positive density on B , we naturally have the notion of L p (B). Choosing a boundary deening function t, the space L p (B) consists of all measurable functions u on B such that Z ju(y)j p t n (y)dd(y) < 1: We introduce a class of weighted Mellin L p-Sobolev spaces H s;; p (B), s; 2 R, 1 < p < 1. For s 2 N they are easily described as the set of all u 2 H s p;loc (int B), for which, in local coordinates on 0; 1) X, t (n+1)=2? (t@ t) k @ x u(t; x) 2 L p (dt t dx); 8 k + jj s: For p = 2 we recover the notation used by Schulze, cf. 30]. Note that L p (B) coincides with H 0;;p p (B) for p = (n + 1)(1=2 ? 1=p). On B we consider the space 2R C (B ; g) of cone pseudodiierential operators as introduced by Schulze (the so-called`weight-datum' g encodes information on the used). An operator A 2 C (B ; g) induces a continuous mapping