We study the de Rham complex of relative differential forms on compact manifolds
with boundary. Chain homotopies for this complex are highly nonunique, 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 Bogovskiĭ, 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.
Keywords
chain homotopy, b-pseudodifferential operator, de Rham
complex, linearized constraint equations of general
relativity
