Acta mathematica scientia,Series A ›› 2025, Vol. 45 ›› Issue (4): 1161-1170.

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A Toeplitz-Type Operator on Hardy Space $H^1(\mathbb{B}_{n})$

Wen Xinqi1,*(),Yuan Cheng2()   

  1. 1Department of Mathematics, Taiyuan University, Taiyuan 030032
    2School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520
  • Received:2025-02-14 Revised:2025-04-14 Online:2025-08-26 Published:2025-08-01
  • Supported by:
    Taiyuan University Youth Research Project(24TYQN20)

Abstract:

This paper investigates the boundedness of a Toeplitz operator $Q_\mu$ acting on the Hardy space $H^1(\mathbb{B}_{n})$. Let $\mu$ be a positive Borel measure on $\mathbb{B}_{n}$ and $0. The main results are following

1. If $\mu$ is a $(1,1)$-logarithmic Carleson measure, then $Q_\mu: H^1(\mathbb{B}_{n})\to H^1(\mathbb{B}_{n})$ is bounded;

2. If $Q_\mu: H^1(\mathbb{B}_{n})\to H^1(\mathbb{B}_{n})$ is bounded, then $\mu$ is a Carleson measure;

3. $Q_\mu: H^p(\mathbb{B}_{n})\to H^q(\mathbb{B}_{n})$ is bounded if and only if $\mu$ is a $(1+\frac1p-\frac1q)$-Carleson measure.

Key words: Hardy spaces, Toeplitz-type operators, Carleson measures

CLC Number: 

  • O174.56
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