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^{-iz}(\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.
