Given a domain $\Omega $ of class $C^{k,1}$, $k\in \Bbb 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 $(\partial- {\partial x_n})\alpha (x',0)= - N(x')$ 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 develop a theory of removable singularities for the weighted Bergman space ${\mathcal A}^p_\mu (\Omega )=\lbrace f \text{analytic} \text{in} \Omega \: \int _\Omega |f|^p \mathrm{d}\mu < \infty \rbrace $, where $\mu $ is a Radon measure on $\mathbb{C}$. The set $A$ is weakly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) \subset \text{Hol}(\Omega )$, and strongly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) = {\mathcal A}^p_\mu (\Omega )$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathop {\mathrm BMO}$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if $\mu $ is absolutely continuous with respect to the Lebesgue measure $m$, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When $\mathrm{d}\mu = w\mathrm{d}m$ and $w$ is a Muckenhoupt $A_p$ weight, $1<p<\infty $, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent $p^{\prime }=p/(p-1)$ and the dual weight $w^{\prime }=w^{1/(1-p)}$.