On the norm of the weighted Berezin transform

2017/11/13 by Petar Melentijević, Melentijević, Petar
Mathematics · #Advanced Harmonic Analysis Research #Complex Variables #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1711.04751

openalex publication_date 2017/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a weighted Berezin transform: B_α : L^∞ (\mathbbBⁿ) → B, αgt;-1, defined, for f ∈ L^∞ ( \mathbbBⁿ ) and z ∈ \mathbbBⁿ, by (B_αf) (z) = c_α∫_\mathbbBⁿ \frac( 1-|z|² )ⁿ⁺¹|1 - ⟨ z, w ⟩|²ⁿ⁺² f(w) ( 1-|w|² )^α d v(w), where c_α = (Γ(α+n+1))/(Γ(α+1)πⁿ) , v is the Lebesque measure and 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≤α≤ 2n+3.

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