A support preserving homotopy for the de Rham complex with boundary decay estimates

TL;DR

This paper constructs a boundary-support-preserving homotopy for the de Rham complex with optimal decay estimates, leading to improved right inverses of divergence operators with support and decay control.

Contribution

It introduces a new homotopy with support propagation properties and decay estimates, utilizing b-pseudodifferential operators, enhancing divergence inverse constructions.

Findings

Supports boundary decay estimates for the de Rham complex

Provides a support-preserving right inverse of divergence on Euclidean space

Extends to divergence of symmetric traceless matrices in three dimensions

Abstract

We study the de Rham complex of relative differential forms on compact manifolds with boundary. Chain homotopies for this complex are highly non-unique, and different homotopies can have different analytic properties, particularly near the boundary. We construct a chain homotopy that has desirable support propagation properties, and that satisfies estimates relative to weighted Sobolev norms, where the weights measure decay at the boundary. The estimates are optimal given the homogeneity properties of the de Rham differential under boundary dilation, and are obtained by showing that the homotopy is a b-pseudodifferential operator. As a corollary we obtain a right inverse of the divergence operator on Euclidean space that preserves support on large balls around the origin, and satisfies estimates that measure decay at infinity. Such a support preserving right inverse was constructed before by Bogovskii, but its mapping properties are not optimal with respect to decay. As a further corollary, in three dimensions we obtain a right inverse of the divergence operator on symmetric traceless matrices, and therefore of the linearized constraint operator of general relativity about flat space.