Characterizations of Adjoint Sobolev Embedding Operators for Inverse Problems

We consider the Sobolev embedding operator $E_s : H^s(\Omega) \to L_2(\Omega)$ and its role in the solution of inverse problems. In particular, we collect various properties and representations of its adjoint operator $E_s^*$, which is a common component in both iterative and variational regularization methods. These include e.g. variational representations and connections to boundary value problems, Fourier and wavelet representations, as well as connections to spatial filters. While many of these results are already known to researchers from different fields, an overview or reference work is still missing. Hence, in this paper we aim to fill this gap, providing a collection of representations of $E_s^*$ which can serve both as a reference as well as a useful guide for its efficient numerical implementation in practice.