On the Forelli-Rudin projection theorem

Motivated by the Forelli--Rudin projection theorem we give in this paper a criterion for boundedness of an integral operator on weighted Lebesgue spaces in the interval $(0,1)$. We also calculate the precise norm of this integral operator. This is the content of the first part of the paper. In the second part, as applications, we give some results concerning the Bergman projection and the Berezin transform. We derive a generalization of the Dostani\'{c} result on the norm of the Berezin transform acting on Lebesgue spaces over the unit ball in $\mathbf{C}^n$.