Regularity of the inverse mapping in Banach function spaces

Abstract

We study the regularity properties of the inverse of a bilipschitz mapping f belonging to WmXloc$W^m X_{\mathrm{loc}}$, where X is an arbitrary Banach function space. Namely, we prove that the inverse mapping f−1$f^{-1}$ is also in WmXloc$W^m X_{\mathrm{loc}}$. Furthermore, the paper shows that the class of bilipschitz mappings in WmXloc$W^m X_{\mathrm{loc}}$ is closed with respect to composition and multiplication.