  In this paper we solve a problem posed by H. Bommier-Hato, M. Engli\v{s} and
E.H. Youssfi in [3] on the boundedness of the Bergman-type projections in
generalized Fock spaces. It will be a consequence of two facts: a full
description of the embeddings between generalized Fock-Sobolev spaces and a
complete characterization of the boundedness of the above Bergman type
projections between weighted $L^p$-spaces related to generalized Fock-Sobolev
spaces.
