Iterative structures on singular manifolds (Q2825872)

The survey under review provides thorough exposition of an iterative approach in the analysis of partial differential equations on manifolds with singularities. The author focuses on operators on manifolds with higher (stratified) singularities in connection with principal symbolic hierarchies, quantization, actions in weighted distribution spaces, ellipticity and parametrices. The iterative methods presented in the article have their origins in the theory of elliptic boundary value problems. Precisely, a two-component principal symbolic hierarchy appears, consisting of the interior symbol and of the boundary symbol that encodes the ellipticity of the boundary conditions. Assume the underlying space \(M\) is a manifold with singularities of order \(k\in\mathbb{N},\) that is, \(k+1\) layers start from the regular, full dimensional part to the \(k\)-th order singular part. Then the principal symbolic hierarchy of an operator \(A\) on \(M\) is given by a sequence of \(k+1\) symbols that determines ellipticity and solvability under additional edge conditions, playing the role of boundary conditions. The operators appearing here are corner-degenerate in stretched representation, and are studied in connection with a suitable quantization which makes them elements of algebras of pseudo-differential operators, controlled both by principal and by complete symbolic hierarchies. In the exposition, the author recalls the category of stratified spaces and introduces the class of corner-degenerate operators, their symbolic hierarchies and the class of attached Sobolev spaces. Detailed description is given of the cone algebra, the edge algebra and the Shapiro-Lopatinskij edge-ellipticity. Further on, the iterated Mellin quantization is presented and Toeplitz edge problems are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 1304.58001].