Regularity of Maps between Sobolev Spaces

TL;DR

This paper demonstrates that equivariance under diffeomorphisms enables a trade-off between the regularity of maps between Sobolev spaces and their images, impacting the analysis of geodesic boundary value problems.

Contribution

It establishes a regularity trade-off for equivariant Sobolev space maps, facilitating analysis of geodesic boundary problems on diffeomorphism groups.

Findings

Regularity of Sobolev maps can be reduced by exploiting equivariance.

The result applies to maps on $ ext{Diff}$ groups and curve spaces.

Improves understanding of geodesic boundary value problems.

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 $0 \leq l \leq k$, 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.