Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916809285828608 |
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| author | Wang, Liyi Ye, Shanli |
| author_facet | Wang, Liyi Ye, Shanli |
| contents | Let $μ$ be a positive Borel measure on the interval $[0,1)$. For $α>0$, the generalized Hankel matrix $\mathcal{H}_{μ, α}=(μ_{n, k, α})_{n, k \geq 0}$ with entries $μ_{n, k, α}=\int_{[0,1)} \frac{Γ(n+α)}{n ! Γ(α)} t^{n+k} \mathrm{d}μ(t)$ induces formally the operator \begin{equation*} \mathcal{H}_{μ, α}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n, k, α} a_k\right) z^n \end{equation*} on the space of all analytic function $f(z)=\sum_{k=0}^{\infty} a_{k} z^{k}$ in the unit disk $\mathbb{D}$. In this paper, we characterize the measures $μ$ for which $\mathcal{H}_{μ, α}(f)$ is well defined on the Hardy spaces $H^p(0<p<\infty)$ and satisfies $\mathcal{H}_{μ, α}(f)(z)=\int_{[0,1)} \frac{f(t)}{(1-t z)^α} \mathrm{d} μ(t)$. Among these measures, we further describe those for which $\mathcal{H}_{μ, α}(α>1)$ is a bounded (resp., compact) operator from the Hardy spaces $H^p(0<p<\infty)$ into the weighted Bergman spaces $A_{α-2}^q $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19338 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces Wang, Liyi Ye, Shanli Complex Variables Functional Analysis 47B38, 30H20, 30H30 Let $μ$ be a positive Borel measure on the interval $[0,1)$. For $α>0$, the generalized Hankel matrix $\mathcal{H}_{μ, α}=(μ_{n, k, α})_{n, k \geq 0}$ with entries $μ_{n, k, α}=\int_{[0,1)} \frac{Γ(n+α)}{n ! Γ(α)} t^{n+k} \mathrm{d}μ(t)$ induces formally the operator \begin{equation*} \mathcal{H}_{μ, α}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n, k, α} a_k\right) z^n \end{equation*} on the space of all analytic function $f(z)=\sum_{k=0}^{\infty} a_{k} z^{k}$ in the unit disk $\mathbb{D}$. In this paper, we characterize the measures $μ$ for which $\mathcal{H}_{μ, α}(f)$ is well defined on the Hardy spaces $H^p(0<p<\infty)$ and satisfies $\mathcal{H}_{μ, α}(f)(z)=\int_{[0,1)} \frac{f(t)}{(1-t z)^α} \mathrm{d} μ(t)$. Among these measures, we further describe those for which $\mathcal{H}_{μ, α}(α>1)$ is a bounded (resp., compact) operator from the Hardy spaces $H^p(0<p<\infty)$ into the weighted Bergman spaces $A_{α-2}^q $. |
| title | Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces |
| topic | Complex Variables Functional Analysis 47B38, 30H20, 30H30 |
| url | https://arxiv.org/abs/2506.19338 |