Polynomial differentiation composition operators from weighted Dirichlet spaces to weighted Bloch spaces

Jing An; Zhi-Jie Jiang; Jing An; Zhi-Jie Jiang

doi:10.3934/era.2026227

Electronic Research Archive

2026, Volume 34, Issue 8: 5119-5143. doi: 10.3934/era.2026227

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Research article

Polynomial differentiation composition operators from weighted Dirichlet spaces to weighted Bloch spaces

Jing An ¹,
Zhi-Jie Jiang ^{1,2
,
,}

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

Received: 10 March 2026 Revised: 10 May 2026 Accepted: 01 June 2026 Published: 15 June 2026

Let $ H({\mathbb D}) $ be the class of all analytic functions on $ \mathbb{D} $, $ {\varphi} $ be an analytic self-map of $ {\mathbb D} $, $ u_0, u_1, \ldots, u_n\in H({\mathbb D}) $, and $ \vec{u} = (u_0, u_1, \ldots, u_{n}) $. In this paper, we studied the operators
$ \begin{align*} P^{n}_{\vec{u}, {\varphi}} = \sum\limits_{j = 0}^{n}D^{j}_{u_j, {\varphi}}, \end{align*} $
where $ D^{j}_{u_j, {\varphi}} $ is the generalized weighted composition operator
$ D^{j}_{u_j, {\varphi}}f = u_j\cdot f^{(j)}\circ{\varphi}, \, \, \, f\in H( {\mathbb D}). $
To be precise, we characterized the boundedness and compactness of the operators $ P^{n}_{\vec{u}, {\varphi}} $ acting from the weighted Dirichlet spaces $ \mathscr{D}^ p_{\alpha} $ to the weighted Bloch spaces for $ p\geq 1 $, and we also calculated the essential norm of such operators for $ p > 1 $.
- weighted Dirichlet space,
- weighted Bloch space,
- polynomial differentiation composition operator,
- boundedness,
- essential norm,
- compactness
Citation: Jing An, Zhi-Jie Jiang. Polynomial differentiation composition operators from weighted Dirichlet spaces to weighted Bloch spaces[J]. Electronic Research Archive, 2026, 34(8): 5119-5143. doi: 10.3934/era.2026227

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Abstract

Let $ H({\mathbb D}) $ be the class of all analytic functions on $ \mathbb{D} $, $ {\varphi} $ be an analytic self-map of $ {\mathbb D} $, $ u_0, u_1, \ldots, u_n\in H({\mathbb D}) $, and $ \vec{u} = (u_0, u_1, \ldots, u_{n}) $. In this paper, we studied the operators

$ \begin{align*} P^{n}_{\vec{u}, {\varphi}} = \sum\limits_{j = 0}^{n}D^{j}_{u_j, {\varphi}}, \end{align*} $

where $ D^{j}_{u_j, {\varphi}} $ is the generalized weighted composition operator

$ D^{j}_{u_j, {\varphi}}f = u_j\cdot f^{(j)}\circ{\varphi}, \, \, \, f\in H( {\mathbb D}). $

To be precise, we characterized the boundedness and compactness of the operators $ P^{n}_{\vec{u}, {\varphi}} $ acting from the weighted Dirichlet spaces $ \mathscr{D}^ p_{\alpha} $ to the weighted Bloch spaces for $ p\geq 1 $, and we also calculated the essential norm of such operators for $ p > 1 $.

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