Sobolev Spaces are useful for the formulation of Inverse Problems as well as the incorporation of, e.g., smoothness properties or statistics of their solution. As a result, the Sobolev embedding operator and its adjoint are common components in both iterative and variational regularization methods for the computation of a solution. However, while the embedding operator itself is trivial, its adjoint is typically not, and the study of its properties and different representations is of importance both theoretically and practically. Hence, in this talk we will present different characterizations of the adjoint embedding operators and their use in standard Tomography, Atmospheric Tomography and Photoacoustic Tomography.

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