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Berezin transforms on modified Bergman spaces

Noureddine Ghiloufi, Safa Snoun

Abstract

In this paper we study the continuity of the Berezin transform on modified Bergman spaces and we establish a Lipschitz estimate in terms of the Bergman-Poincaré metric.

Berezin transforms on modified Bergman spaces

Abstract

In this paper we study the continuity of the Berezin transform on modified Bergman spaces and we establish a Lipschitz estimate in terms of the Bergman-Poincaré metric.
Paper Structure (8 sections, 8 theorems, 49 equations)

This paper contains 8 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Berezin transform
  4. Definition and properties of $\mathbb B_{\alpha,\beta}$
  5. Proof of Theorem \ref{['th1']}
  6. Bergman-Poincaré metric and application
  7. BMO in Bergman-Poincaré metric
  8. Proof of Theorem \ref{['th2']}

Key Result

Theorem 1

Let $-1<\alpha,a,b<+\infty,\ -1<\beta\leq0$ and $1\leq p<+\infty$. Then the Berezin transform $\mathbb B_{\alpha,\beta}:L^p(\mathbb D,\mu_{a,b})\longrightarrow L^p(\mathbb D,\mu_{a,b})$ is bounded if and only if

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • ...and 8 more