On the $p$-norm of the Berezin transform

Abstract

In this short note, the norm of Berezin transform, acting on $L^{p}(\mathbb{B}_{n})$, is determined to be

\[\bigl\|B_{\mathbb{B} _{n}}:L^{p}(\mathbb{B}_{n})\to L^{p}(\mathbb{B}_{n})\bigr\|=\frac{1}{p}\prod_{k=1}^{n}\biggl(1+\frac{1}{kp}\biggr)\frac{\pi}{\sin(\pi/p)}.\]

This extends a result of Dostanić (J. Anal. Math. 104 (2008) 13–23) to several complex variables.