Operators on Corner Manifolds with Exit to Infinity

Abstract

We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ_ ψ(A), σ_^(A), σ_^(A)), where σ_ ψ is the interior symbol and σ^(A)(y, η), (y, η) ∈ T∗Y \0, the (operator-valued) edge symbol of ‘first generation̻, cf. [1]. The novelty here is the edge symbol σ_^ of ‘second generation’, parametrised by (z, ζ) ∈ T∗Z \ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.