  We study the weak regularity of mappings inverse to weighted Sobolev
homeomorphisms $\varphi:\Omega\to\widetilde{\Omega}$, where $\Omega$ and
$\widetilde{\Omega}$ are domains in $\mathbb R^n$. Using the weak regularity of
inverse mappings we obtain the composition duality property of composition
operators on weighted Sobolev spaces.
