On the Forelli–Rudin Projection Theorem

Abstract

Motivated by the Forelli–Rudin projection theorem we give in this paper a criteria for boundedness of an integral operator on Lebesgue spaces in the interval (0, 1). We also give the precise norm of this integral operator. As a consequence, one can derive a generalization of the Dostanić result concerning the norm of the Berezin transform ${\mathfrak{B}}$ acting on the Lebesgue space L ^p(B) of the unit ball in ${\mathbb{C}^n}$ which says that

$$\|\mathfrak{B} : L^p(B) \rightarrow L^p(B) \| = \left\{\prod_{k=1}^n \left(1 + \frac 1{ k\, p} \right) \right\} \frac {\frac \pi p}{\sin \frac \pi p}$$

for any real p greater then 1. The result belong to Dostanić in the case n = 1.