Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball

Cheng-shi Huang; Zhi-jie Jiang; Yan-fu Xue; Cheng-shi Huang; Zhi-jie Jiang; Yan-fu Xue

doi:10.3934/math.20221001

AIMS Mathematics

2022, Volume 7, Issue 10: 18194-18217. doi: 10.3934/math.20221001

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Research article

Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball

1.
School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China
2.
South Sichuan Center for Applied Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China

Received: 17 May 2022 Revised: 12 July 2022 Accepted: 26 July 2022 Published: 11 August 2022
MSC : 30H05, 47B33, 47B37, 47B38

Let $ u_{j} $ be the holomorphic functions on the open unit ball $ \mathbb{B} $ in $ \mathbb{C}^{n} $, $ j = \overline{0, m} $, $ \varphi $ a holomorphic self-map of $ \mathbb{B} $, and $ \Re^{j} $ the $ j $th iterated radial derivative operator. In this paper, the boundedness and compactness of the sum operator $ \mathfrak{S}^m_{\vec{u}, \varphi} = \sum_{j = 0}^m M_{u_j}C_\varphi\Re^j $ from the mixed-norm space $ H(p, q, \phi) $, where $ 0 < p, q < +\infty $, and $ \phi $ is normal, to the weighted-type space $ H^\infty_\mu $ are characterized. For the mixed-norm space $ H(p, q, \phi) $, $ 1\leq p < +\infty $, $ 1 < q < +\infty $, the essential norm estimate of the operator is given, and the Hilbert-Schmidt norm of the operator on the weighted Bergman space $ A^2_\alpha $ is also calculated.
- product-type operator,
- boundedness,
- compactness,
- mixed-norm space,
- weighted-type space,
- essential norm,
- Hilbert-Schmidt norm
Citation: Cheng-shi Huang, Zhi-jie Jiang, Yan-fu Xue. Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball[J]. AIMS Mathematics, 2022, 7(10): 18194-18217. doi: 10.3934/math.20221001

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Abstract

Let $ u_{j} $ be the holomorphic functions on the open unit ball $ \mathbb{B} $ in $ \mathbb{C}^{n} $, $ j = \overline{0, m} $, $ \varphi $ a holomorphic self-map of $ \mathbb{B} $, and $ \Re^{j} $ the $ j $th iterated radial derivative operator. In this paper, the boundedness and compactness of the sum operator $ \mathfrak{S}^m_{\vec{u}, \varphi} = \sum_{j = 0}^m M_{u_j}C_\varphi\Re^j $ from the mixed-norm space $ H(p, q, \phi) $, where $ 0 < p, q < +\infty $, and $ \phi $ is normal, to the weighted-type space $ H^\infty_\mu $ are characterized. For the mixed-norm space $ H(p, q, \phi) $, $ 1\leq p < +\infty $, $ 1 < q < +\infty $, the essential norm estimate of the operator is given, and the Hilbert-Schmidt norm of the operator on the weighted Bergman space $ A^2_\alpha $ is also calculated.

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