  We study underdetermined elliptic linear partial differential operators $P$
on asymptotically Euclidean manifolds, such as the divergence operator on
1-forms or symmetric 2-tensors. Suitably interpreted, these are instances of
(weighted) totally characteristic differential operators on a compact manifold
with boundary whose principal symbols are surjective but not injective. We
study the equation $P u=f$ when $f$ has a generalized Taylor expansion at
$r=\infty$, that is, a full asymptotic expansion into terms with radial
dependence $r^{-i z}(\log r)^k$ with $(z,k)\in\mathbb{C}\times\mathbb{N}_0$ up
to rapidly decaying remainders. We construct a solution $u$ whose asymptotic
behavior at $r=\infty$ is optimal in that the index set of exponents $(z,k)$
arising in its asymptotic expansion is as small as possible. On the flipside,
we show that there is an infinite-dimensional nullspace of $P$ consisting of
smooth tensors whose expansions at $r=\infty$ contain nonzero terms $r^{-i
z}(\log r)^k$ for any desired index set of
$(z,k)\in\mathbb{C}\times\mathbb{N}_0$.
  Applications include sharp solvability results for the divergence equation on
1-forms or symmetric 2-tensors on asymptotically Euclidean spaces, as well as a
regularity improvement in a gluing construction for the constraint equations in
general relativity recently introduced by the author.
