Boundary value problems for 0-elliptic operators

TL;DR

This paper develops a new symbolic 0-calculus for boundary value problems involving 0-elliptic operators on manifolds with boundary, enabling construction of parametrices with elliptic boundary conditions.

Contribution

It introduces the symbolic 0-calculus, extending Mazzeo--Melrose's 0-calculus to handle boundary value problems for 0-elliptic operators.

Findings

Constructed left and right parametrices for 0-elliptic operators with boundary conditions.

Developed a new calculus of pseudodifferential operators called the symbolic 0-calculus.

Enabled analysis of boundary value problems for operators like Hodge Laplacians and Dirac operators.

Abstract

Let $X$ be a manifold with boundary, and let $L$ be a 0-elliptic operator on X which is semi-Fredholm essentially surjective with infinite-dimensional kernel. Examples include Hodge Laplacians and Dirac operators on conformally compact manifolds. We construct left and right parametrices for L when supplemented with appropriate elliptic boundary conditions. The construction relies on a new calculus of pseudodifferential operators on functions over both $X$ and $\partial X$, which we call the "symbolic 0-calculus". This new calculus supplements the ordinary 0-calculus of Mazzeo--Melrose, enabling it to handle boundary value problems. In the original 0-calculus, operators are characterized as polyhomogeneous right densities on a blow-up of $X^2$. By contrast, operators in the symbolic 0-calculus are characterized (locally near each point of the boundary of the diagonal) as quantizations of polyhomogeneous symbols on appropriate blown-up model spaces.