The Weak Inverse Mapping Theorem

Abstract

We prove that if a bilipschitz mapping $f$ is in $W_{\mathrm{loc}}^{m,p}({\mathbb{R}}^n,{\mathbb{R}}^n)$ then the inverse $f^{-1}$ is also a $W_{\mathrm{loc}}^{m,p}$ class mapping. Further we prove that the class of bilipschitz mappings belonging to $W_{\mathrm{loc}}^{m,p}({\mathbb{R}}^n,{\mathbb{R}}^n)$ is closed with respect to composition and multiplication without any restrictions on $m,p\ge 1$. These results can be easily extended to smooth $n$-dimensional Riemannian manifolds and further we prove a form of the implicit function theorem for Sobolev mappings.