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 investigate the steady transport equation $$\lambda z+w·\nabla z+az=f,\lambda >0$$ in various domains (bounded or unbounded) with smooth noncompact boundaries. The functions $w,a$ are supposed to be small in appropriate norms. The solution is studied in spaces of Sobolev type (classical Sobolev spaces, Sobolev spaces with weights, homogeneous Sobolev spaces, dual spaces to Sobolev spaces). The particular stress is put onto the problem to extend the results to as less regular vector fields $w,a$, as possible (conserving the requirement of...