Generalized Hilbert operator acting on Bergman spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916540570402816 |
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| author | Tang, Pengcheng Zhang, Xuejun |
| author_facet | Tang, Pengcheng Zhang, Xuejun |
| contents | Let $μ$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$ and $α>-1$, the generalized integral type Hilbert operator defined as follows:
$$\mathcal{I}_{μ_{α+1}}(f)(z)=\int^1_{0} \frac{f(t)}{(1-tz)^{α+1}}dμ(t), \ \ \ z\in \mathbb{D} .$$ The operator $\mathcal{I}_{μ_{1}}$ has been extensively studied recently. In this paper, we characterize the measures $μ$ for which $\mathcal{I}_{μ_{α+1}}$ is a bounded (resp., compact) operator acting between the Bloch space $\mathcal {B}$ and Bergman space $ A^{p}$, or from $A^{p}(0<p<\infty)$ into $ A^{q}(q\geq 1)$. We also study the analogous problem in Bergman spaces $A^{p}(1 \leq p\leq 2)$. Finally, we determine the Hilbert-Schmidt class on $A^{2}$ for all $α>-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_10747 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generalized Hilbert operator acting on Bergman spaces Tang, Pengcheng Zhang, Xuejun Functional Analysis Complex Variables Let $μ$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$ and $α>-1$, the generalized integral type Hilbert operator defined as follows: $$\mathcal{I}_{μ_{α+1}}(f)(z)=\int^1_{0} \frac{f(t)}{(1-tz)^{α+1}}dμ(t), \ \ \ z\in \mathbb{D} .$$ The operator $\mathcal{I}_{μ_{1}}$ has been extensively studied recently. In this paper, we characterize the measures $μ$ for which $\mathcal{I}_{μ_{α+1}}$ is a bounded (resp., compact) operator acting between the Bloch space $\mathcal {B}$ and Bergman space $ A^{p}$, or from $A^{p}(0<p<\infty)$ into $ A^{q}(q\geq 1)$. We also study the analogous problem in Bergman spaces $A^{p}(1 \leq p\leq 2)$. Finally, we determine the Hilbert-Schmidt class on $A^{2}$ for all $α>-1$. |
| title | Generalized Hilbert operator acting on Bergman spaces |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2208.10747 |