Regularity of Maps between Sobolev Spaces

Abstract

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 $0\le l\le k$, 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.

13 pages