On the norm of the weighted Berezin transform

Abstract

We consider a weighted Berezin transform: $$B_{\alpha }:L^{\infty }({\mathbb{B}}^n)\to \mathcal{B},\alpha >-1,$$ defined, for $f\in L^{\infty }\left({\mathbb{B}}^n\right)$ and $z\in {\mathbb{B}}^n$, by $$(B_{\alpha }f)(z)=c_{\alpha }{\int }_{{\mathbb{B}}^n}\frac{{\left(1-|z{|}^2\right)}^{n+1}}{|1-⟨z,w⟩{|}^{2n+2}}f(w){\left(1-|w{|}^2\right)}^{\alpha }dv(w),$$ 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 $0\le \alpha \le 2n+3.$

20 pages