On the \(p\)-norm of the Berezin transform (Q384318)

Let \(\nu_n\) denote the normalized Lebesgue measure on the unit ball \(B_n\) of \(\mathbb{C}^n\), \(n\geq 1\). Given \(f\in L^1(B_n, \nu_n)\), the Berezin transform \(\mathcal{B}f\) is defined by the formula NEWLINE\[NEWLINE \mathcal{B} f(z)= (1-|z|^2)^{n+1} \int_{B_n} \frac{f(w)\, d\nu_n(w)}{|1-\langle z, w \rangle|^{2(n+1)}}, \quad z\in B_n. NEWLINE\]NEWLINE The authors prove that NEWLINE\[NEWLINE \left\|\mathcal{B}: L^p(B_n)\to L^p(B_n) \right\|= \frac{\pi}{p\sin(\pi/p)}\prod_{k=1}^n \left(1+\frac{1}{kp} \right), \quad 1<p\leq\infty. NEWLINE\]NEWLINE For \(n=1\), the above formula was earlier obtained by \textit{M. Dostanić} [J. Anal. Math. 104, 13--23 (2008; Zbl 1155.47051)].