The Calderón projection: New definition and applications

Abstract

Abstract. We consider an arbitrary linear elliptic first-order differential operator A with smooth coefficients acting between sections of complex vector bundles E, F over a compact smooth manifold M with smooth boundary Σ. We describe the analytic and topological properties of A in a collar neighborhood U of Σ and analyze various ways of writing A U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderón projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderón projection and of well-posed selfadjoint Fredholm extensions under continuous variation of the data.

The Invertible Double of Elliptic Operators

“…That provides a formula for C + in terms of A such that the Lagrangian property (N 0 + ) ⊥ = JN 0 + is implied, see [10,Corollary 12.6], and mapping properties and dependencies on the data become transparent, see [6]. Only recently, the authors of this short review were able to prove similar results for general elliptic differential operators of first order, in collaboration with C. Zhu [7] (see Theorems 3.2, 3.4, and 3.5 below).…”

“…This paper reviews our recent results, obtained jointly with Chaofeng Zhu [7], about basic analytical properties of elliptic operators on compact manifolds with smooth boundary. Furthermore, we outline a research program for confining, respectively closing, the last remaining gaps between the geometric Dirac operator type situation and the general linear elliptic case.…”

“…Continuous variation of P + can also be obtained for general A and general B 0 if the variation is of order < 1 [7,Prop. 7.13].…”

“…Namely, if we restrict to self-adjoint tangential operators B 0 then the Spectral Theorem allows to prove (cf. [7,Prop. 7.15].…”

“…We only give sketches of the proofs and made full-length proofs available at arXiv, Booß, Lesch and Zhu [7].…”