The Calderon Projection: New Definition and Applications

TL;DR

This paper introduces a new, canonical construction of the Calderon projection for elliptic first-order operators on manifolds with boundary, extending its applications to cobordism and boundary condition analysis.

Contribution

It provides a novel, unified definition of the Calderon projection and explores its topological and analytical properties, with applications to boundary value problems and operator extensions.

Findings

Constructed a canonical Calderon projection for elliptic operators.

Generalized the Cobordism Theorem using the Calderon projection.

Identified conditions for continuous variation of boundary projections.

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 N. We describe the analytic and topological properties of A in a collar neighborhood U of N 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 Calderon projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderon projection and of well--posed selfadjoint Fredholm extensions under continuous variation of the data.