  The authors consider a scattering problem for electric potentials that have a
component which is critically singular in the sense of Lebesgue spaces, and a
component given by a measure supported on a compact Lipschitz hypersurface.
They study direct and inverse point-source scattering under the assumptions
that the potentials are real-valued and compactly supported. To solve the
direct scattering problem, the authors introduce two functional spaces ---sort
of Bourgain type spaces--- that allow to refine the classical resolvent
estimates of Agmon and H\"ormander, and Kenig, Ruiz and Sogge. These spaces
seem to be very useful to deal with the critically-singular and $\delta$-shell
components of the potentials at the same time. Furthermore, these spaces and
their corresponding resolvent estimates turn out to have a strong connection
with the estimates for the conjugated Laplacian used in the context of the
inverse Calder\'on problem. In fact, the authors derive the classical estimates
by Sylvester and Uhlmann, and the more recent ones by Haberman and Tataru after
some embedding properties of these new spaces. Regarding the inverse scattering
problem,the authors prove uniqueness for the potentials from point-source
scattering data at fix energy. To address the question of uniqueness the
authors combine some of the most advanced techniques in the construction of
complex geometrical optics solutions.
