Elliptic systems of variable order

TL;DR

This paper develops a comprehensive pseudodifferential calculus for elliptic wedge operators on manifolds with boundary, incorporating vector bundle actions, Sobolev spaces, and an index theorem, extending classical boundary value problem theory.

Contribution

It introduces a full pseudodifferential operator calculus for elliptic wedge operators, including symbols, Sobolev spaces, and an index theorem, in a general boundary value problem setting.

Findings

Defined symbols for wedge pseudodifferential operators using group actions

Presented ancillary Sobolev spaces for elliptic wedge operators

Established an index theorem for elliptic elements in the calculus

Abstract

The general theory of boundary value problems for linear elliptic wedge operators (on smooth manifolds with boundary) leads naturally, even in the scalar case, to the need to consider vector bundles over the boundary together with general smooth fiberwise multiplicative group actions. These actions, essentially trivial (and therefore invisible) in the case of regular boundary value problems, are intimately connected with what passes for Poisson and trace operators, and to pseudodifferential boundary conditions in the more general situation. Here the part of the theory pertaining pseudodifferential operators is presented in its entirety. The symbols for the latter operators are defined with the aid of an intertwining of the actions. Also presented here are the ancillary Sobolev spaces, an index theorem for the elliptic elements of the pseudodifferential calculus, and the essential ingredients of the APS boundary condition in the more general theory.