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Hausdorff operators on weighted mixed norm Fock spaces

Yongqing Liu

Abstract

In this paper, we study Hausdorff operator $\mathcal{H}_μ$ on a large class of weighted mixed norm Fock spaces $F_φ^{p,q}$ for $1\leq p,q\leq\infty$. The boundedness and compactness of $\mathcal{H}_μ$ on $F_φ^{p,q}$ are characterized. As applications, we give when Hausdorff operator on $F_φ^{p,q}$ is power bounded or uniformly mean ergodic.

Hausdorff operators on weighted mixed norm Fock spaces

Abstract

In this paper, we study Hausdorff operator on a large class of weighted mixed norm Fock spaces for . The boundedness and compactness of on are characterized. As applications, we give when Hausdorff operator on is power bounded or uniformly mean ergodic.

Paper Structure

This paper contains 4 sections, 11 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Boundedness and compactness of $\mathcal{H}_\mu$
  3. Power boundedness and uniformly mean ergodicity of $\mathcal{H}_\mu$
  4. Further comments

Key Result

Proposition 1

Let $1\leq p,q \leq\infty$ and $\phi$ be a weight. The dilation operator $\mathcal{D}_t$ is bounded on $F_\phi^{p,q}$ if and only if $t\geq1$. Moreover, we have

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • Theorem 2
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['boundedness']}
  • Corollary 4
  • Theorem 5
  • Lemma 6: ACFPP Lemma 4.10
  • Lemma 7
  • ...and 10 more