Boundedness of the Bergman projection on generalized Fock-Sobolev spaces on ${\mathbb{C}}^n$

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.