Operators on manifolds with conical singularities
The authors consider general nature of pseudo-differential algebras on a manifold with conical or edge singularities of order 1. The presence of symbolic hierarchies is a new aspect, compared with the smooth case. Ellipticity of an operator \(A\) on such space is a bijectivity condition on a principal symbolic hierarchy \((\sigma_0,\sigma_1)\) where \(\sigma_0\) is the standard degenerate homogeneous principal symbol, and \(\sigma_1\) is the so-called conormal symbol. They show the existence of a smoothing Mellin operator \(F\) belonging to the cone algebra such that \(A+F\) is also elliptic with respect to \(\sigma_1\) for a prescribed weight \(\gamma\in\mathbb{R}\). This result considerably simplifies the parametrix construction in the cone algebra.
operators on manifolds with conical singularities
parametrices and invertibility within the cone calculus
scientific article; zbMATH DE number 5809988