On the norm of the weighted Berezin transform

Abstract

We consider a weighted Berezin transform: $$ B_{\alpha} : L^{\infty} (\mathbb{B}^n) \to \ \mathcal{B},\quad \alpha>-1,$$ defined, for $f \in L^{\infty} \left( \mathbb{B}^n \right)$ and $z \in \mathbb{B}^n$, by $$(B_\alpha f) (z) = c_\alpha \int_{\mathbb{B}^n} \frac{\left( 1-|z|^2 \right)^{n+1}}{|1 - \langle z, w \rangle|^{2n+2}} f(w) \left( 1-|w|^2 \right)^\alpha \ d v(w),$$ where $c_{\alpha} = \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+1)\pi^n}$ , $v$ is the Lebesque measure and $\mathcal{B}$ is a Bloch-type space. We prove that $B_{\alpha}$ is bounded iff $\alpha>0$ and give the exact semi-norm of $ B_\alpha$ for $0\leq\alpha \leq 2n+3.$