Kato constants in Riemannian geometry

We derive improvements of Kato's inequality for sections of an irreducible SO(n)- or Spin(n)-bundle v in the category of Riemannian manifolds, under the assumption that the section solves a first-order equivariant injectively elliptic system D. An explicit, general (covering all v and D) formula is given for the improved Kato constant.