Hankel-Type Operator Acting on Hardy Spaces and Weighted Bergman Spaces

IF 0.7 4区 数学 Q2 MATHEMATICS
Zhihui Zhou
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引用次数: 0

Abstract

Inspired by Xiao’s work about the Hankel measures for the weighted Bergman spaces, in this paper, if \(\beta >0\) and the measure \(\mu \) is a complex Borel measure on the unit disk \({\mathbb {D}}\), we define the Hankel type operator \(K_{\mu ,\beta }\) by

$$\begin{aligned} K_{\mu ,\beta }:~f\longmapsto \int _{{\mathbb {D}}}(1-wz)^{-(\beta )}f(w)d\mu (w). \end{aligned}$$

The operator itself has been widely studied when \(\mu \) is a positive Borel measure supported on the interval [0, 1). We study the boundedness of \(K_{\mu ,1}\) acting on Hardy spaces and the boundedness of \(K_{\mu ,\alpha }\), \(\alpha >1\) acting on weighted Bergman spaces. Then we raise and answer some questions about the boundedness of those operators. Also, we find some special measures \(\mu 's\) such that s-Hankel measure is equal to s-Carleson measure.

作用于哈代空间和加权伯格曼空间的汉克尔型算子
受肖恩关于加权伯格曼空间的汉克尔度量的启发,在本文中,如果 \(\beta >0\) 和度量 \(\mu \) 是单位盘 \({\mathbb{D}}\)上的复 Borel 度量,我们通过 $$\begin{aligned}定义汉克尔型算子 \(K_{\mu ,\beta }\)K_{{mu ,\beta }:~f\longmapsto int _{{\mathbb {D}}}(1-wz)^{-(\beta )}f(w)d\mu (w).\end{aligned}$$当 \(\mu \)是一个支持区间 [0, 1) 的正波尔度量时,算子本身已经被广泛研究。我们研究了作用于哈代空间的 \(K_{\mu ,1}\) 的有界性,以及作用于加权伯格曼空间的 \(K_{\mu ,\alpha }\), \(\alpha >1\) 的有界性。然后,我们提出并回答了关于这些算子有界性的一些问题。此外,我们还发现了一些特殊的度量 \(\mu 's\) ,使得s-Hankel度量等于s-Carleson度量。
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来源期刊
CiteScore
1.20
自引率
12.50%
发文量
107
审稿时长
3 months
期刊介绍: Complex Analysis and Operator Theory (CAOT) is devoted to the publication of current research developments in the closely related fields of complex analysis and operator theory as well as in applications to system theory, harmonic analysis, probability, statistics, learning theory, mathematical physics and other related fields. Articles using the theory of reproducing kernel spaces are in particular welcomed.
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