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On the norms of the multiplication operators between weighted Bergman spaces

Jianjun Jin

Abstract

In this paper, we investigate the norms of multiplication operators acting between weighted Bergman spaces. First, we provide a proof for a norm estimate claim previously announced in our recent paper. Second, we establish a sharp norm estimate for certain special multiplication operators between weighted Bergman spaces, a result that appears to be new in the literature.

On the norms of the multiplication operators between weighted Bergman spaces

Abstract

In this paper, we investigate the norms of multiplication operators acting between weighted Bergman spaces. First, we provide a proof for a norm estimate claim previously announced in our recent paper. Second, we establish a sharp norm estimate for certain special multiplication operators between weighted Bergman spaces, a result that appears to be new in the literature.
Paper Structure (4 sections, 11 theorems, 171 equations)

This paper contains 4 sections, 11 theorems, 171 equations.

Table of Contents

  1. Introduction and main results
  2. Some Auxiliary Results
  3. Proof of Theorem \ref{['m-1']}
  4. Proof of Theorem \ref{['m-2']}

Key Result

Theorem 1.1

Let $\beta>\alpha>-1$. The multiplication operator $M_g$ is bounded from ${\bf A}_{\alpha}^2$ to ${\bf A}_{\beta}^2$ if and only if $g$ belongs to $\mathcal{B}_{\gamma}$ with $\gamma=(\beta-\alpha)/2.$

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • Lemma 2.1
  • Remark 2.2
  • ...and 18 more