Boundary Value Problems and Edge Pseudo-Differential Operators

Abstract

The analysis of pseudo-differential operators on a closed compact C∞ manifold (in its standard form) allows the construction of parametrices of elliptic operators by inverting local symbols and forming the associated operators. Elliptic regularity and the Fredholm property of elliptic operators in Sobolev spaces are consequences of the basic calculus of pseudo-differential operators. It is well-known how the interplay between symbolic and operator level, together with homotopy and operator algebra aspects, are involved in the index theory in K-theoretic terms, cf. Atiyah, Singer [2], in the program to express the index by analytical formulas, cf. Fedosov [9] or in other strategies for analyzing and interpreting the index, e.g., by the heat kernel asymptotics. For interesting classes of singular or non-compact manifolds, essential problems like adequate operator algebras with symbolic structures, the definition of ellipticity, and index theory, are unsolved.