dorsal/arxiv
View SchemaOperators of Hilbert type acting on some spaces of analytic functions
| Authors | Pengcheng Tang |
|---|---|
| Categories | |
| ArXiv ID | 2601.08473vv1 |
| URL | https://arxiv.org/abs/2601.08473 |
| License | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Abstract
Let $H(\mathbb{D})$ be the space of all analytic functions in the unit disc $\mathbb{D}$. For $g\in H(\mathbb{D})$, the generalized Hilbert operator $\mathcal{H}_{g}$ is defined by $$\mathcal{H}_{g}(f)(z)=\int_{0}^{1}f(t)g'(tz)dt, \ \ z\in \mathbb{D}, f\in H(\mathbb{D}).$$ In this paper, we study the operator $\mathcal{H}_{g}$ acting on some spaces of analytic functions in $\mathbb{D}$. Specifically, we give a complete characterization of those $g\in H(\mathbb{D})$ for which the operator $\mathcal{H}_{g}$ is bounded (resp. compact) from the Dirichlet space $\mathcal{D}^{2}_{\alpha}$ to $\mathcal{D}^{2}_{\beta}$ for all possible indicators $\alpha,\beta \in \mathbb{R}$. We also study the action of the operator $\mathcal{H}_{g}$ on the space of bounded analytic functions $H^{\infty}$, which generalizes the known results for the classical Hilbert operator $\mathcal {H}$ acting on $H^{\infty}$. In particular, we consider the boundedness of the operator $\mathcal{H}_{g}$ with a symbol of non-negative Taylor coefficients, acting on logarithmic Bloch spaces and on Korenblum spaces. This work generalizes the corresponding results for the classical Hilbert operator.
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"abstract": "Let $H(\\mathbb{D})$ be the space of all analytic functions in the unit disc $\\mathbb{D}$. For $g\\in H(\\mathbb{D})$, the generalized Hilbert operator $\\mathcal{H}_{g}$ is defined by $$\\mathcal{H}_{g}(f)(z)=\\int_{0}^{1}f(t)g\u0027(tz)dt, \\ \\ z\\in \\mathbb{D}, f\\in H(\\mathbb{D}).$$\n In this paper, we study the operator $\\mathcal{H}_{g}$ acting on some spaces of analytic functions in $\\mathbb{D}$. Specifically, we give a complete characterization of those $g\\in H(\\mathbb{D})$ for which the operator $\\mathcal{H}_{g}$ is bounded (resp. compact) from the Dirichlet space $\\mathcal{D}^{2}_{\\alpha}$ to $\\mathcal{D}^{2}_{\\beta}$ for all possible indicators $\\alpha,\\beta \\in \\mathbb{R}$. We also study the action of the operator $\\mathcal{H}_{g}$ on the space of bounded analytic functions $H^{\\infty}$, which generalizes the known results for the classical Hilbert operator $\\mathcal {H}$ acting on $H^{\\infty}$. In particular, we consider the boundedness of the operator $\\mathcal{H}_{g}$ with a symbol of non-negative Taylor coefficients, acting on logarithmic Bloch spaces and on Korenblum spaces. This work generalizes the corresponding results for the classical Hilbert operator.",
"arxiv_id": "2601.08473",
"authors": [
"Pengcheng Tang"
],
"categories": [
"math.FA"
],
"license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
"title": "Operators of Hilbert type acting on some spaces of analytic functions",
"url": "https://arxiv.org/abs/2601.08473",
"version": "v1"
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