Sharp Forelli–Rudin estimates and the norm of the Bergman projection

The purpose of this paper is twofold. We first establish a sharp version of Forelli–Rudin estimates for certain integrals on the ball. Then, as main application of these estimates, we obtain a sharp LpLp-norm estimate for the Bergman projection, which refines a result of K. Zhu as well as gives a negative answer to a question raised by M. Dostanić.