# Regularity of the inverse mapping in Banach function spaces

## TL;DR

This paper investigates the regularity of inverse mappings in Banach function spaces, proving that bilipschitz mappings in certain Sobolev-type spaces have inverses with similar regularity and that these classes are stable under composition and multiplication.

## Contribution

It establishes the regularity of inverse mappings in Banach function spaces and shows stability of bilipschitz classes under composition and multiplication.

## Key findings

- Inverse mappings in $W^m X_{loc}$ are also in $W^m X_{loc}$.
- The class of bilipschitz mappings in $W^m X_{loc}$ is closed under composition.
- Stability of bilipschitz mappings under multiplication.

## Abstract

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