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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 34, Issue 3, 2015, pp. 321–342**

**DOI: 10.4171/ZAA/1542**

Published online: 2015-07-08

The Weak Inverse Mapping Theorem

Daniel Campbell^{[1]}, Stanislav Hencl

^{[2]}and František Konopecký

^{[3]}(1) University of Hradec Králové, Czech Republic

(2) Charles University, Prague, Czech Republic

(3) Charles University, Prague, Czech Republic

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 $m, p \geq 1$. 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.

*Keywords: *Bilipschitz mappings, inverse mapping theorem

Campbell Daniel, Hencl Stanislav, Konopecký František: The Weak Inverse Mapping Theorem. *Z. Anal. Anwend.* 34 (2015), 321-342. doi: 10.4171/ZAA/1542