  Let $F : H^q \to H^q$ be a $C^k$-map between Sobolev spaces, either on
$\mathbb R^d$ or on a compact manifold. We show that equivariance of $F$ under
the diffeomorphism group allows to trade regularity of $F$ as a nonlinear map
for regularity in the image space: for $0 \leq l \leq k$, the map $F: H^{q+l}
\to H^{q+l}$ is well-defined and of class $C^{k-l}$. This result is used to
study the regularity of the geodesic boundary value problem for Sobolev metrics
on the diffeomorphism group and the space of curves.
