  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 $\mathbf{C}^n$.

Complex Variables