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

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