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Given a domain $\Omega$ of class $C^{k,1}$, $k\in \mathbb{N}$, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of $\Omega$ in the sense that $(\partial -\partial x_n)\alpha (x^{\text{'}},0)=-N\left(x^{\text{'}}\right)$ and that still is of class $C^{k,1}$. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to $k$ on domains of class $C^{k,1}$. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.

We consider the problem div u = f in a bounded Lipschitz domain Ω, where f with ${\int }_{\Omega }f=0$ is given. It is shown that the solution u, constructed as in Bogovski’s approach in [1], fulfills estimates in the weighted Sobolev spaces $W_w^{k,q}\left(\Omega \right)$, where the weight function w is in the class of Muckenhoupt weights $A_q$.