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 ...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 Cmodulo ＇lower order terms＇） by a principal symbolic hierarchy σ（A） = （σψ （A）, σ∧ CA）, σ∧ （A））, where σψ is the interior symbol and σ∧（A） （y, η）, （y, η） ∈ T＊Y／0, the Coperator-valued） edge symbol of ＇first generation＇, 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.展开更多
文摘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 Cmodulo ＇lower order terms＇） by a principal symbolic hierarchy σ（A） = （σψ （A）, σ∧ CA）, σ∧ （A））, where σψ is the interior symbol and σ∧（A） （y, η）, （y, η） ∈ T＊Y／0, the Coperator-valued） edge symbol of ＇first generation＇, 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.
