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arXiv · 1711.04751

On the norm of the weighted Berezin transform

Petar Melentijević
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Abstract

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

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Petar Melentijević. 2018-01-23. On the norm of the weighted Berezin transform. https://arxiv.org/abs/1711.04751

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