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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2007.15402 |
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| _version_ | 1866917270948675584 |
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| author | Peláez, José Ángel de la Rosa, Elena |
| author_facet | Peláez, José Ángel de la Rosa, Elena |
| contents | We consider the Hilbert-type operator defined by
$$
H_ω(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^ω_t(u)\,du\right)\,ω(t)dt,$$ where $\{B^ω_ζ\}_{ζ\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_ω$ induced by a radial weight $ω$ in the unit disc $\mathbb{D}$. We prove that $H_ω$ is bounded from $H^\infty$ to the Bloch space if and only if $ω$ belongs to the class $\widehat{\mathcal{D}}$, which consists of radial weights $ω$ satisfying the doubling condition $\sup_{0\le r<1} \frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1ω(s)\,ds}<\infty$. Further, we describe the weights $ω\in \widehat{\mathcal{D}}$ such that $H_ω$ is bounded on the Hardy space $H^1$, and we show that for any $ω\in \widehat{\mathcal{D}}$ and $p\in (1,\infty)$, $H_ω:\,L^p_{[0,1)} \to H^p$ is bounded if and only if the Muckenhoupt type condition
\begin{equation*}
\sup\limits_{0<r<1}\left(1+\int_0^r \frac{1}{\widehatω(t)^p} dt\right)^{\frac{1}{p}}
\left(\int_r^1 ω(t)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty,
\end{equation*} holds. Moreover, we address the analogous question about the action of $H_ω$ on weighted Bergman spaces $A^p_ν$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_15402 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Hilbert-type operator induced by radial weight Peláez, José Ángel de la Rosa, Elena Complex Variables Functional Analysis We consider the Hilbert-type operator defined by $$ H_ω(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^ω_t(u)\,du\right)\,ω(t)dt,$$ where $\{B^ω_ζ\}_{ζ\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_ω$ induced by a radial weight $ω$ in the unit disc $\mathbb{D}$. We prove that $H_ω$ is bounded from $H^\infty$ to the Bloch space if and only if $ω$ belongs to the class $\widehat{\mathcal{D}}$, which consists of radial weights $ω$ satisfying the doubling condition $\sup_{0\le r<1} \frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1ω(s)\,ds}<\infty$. Further, we describe the weights $ω\in \widehat{\mathcal{D}}$ such that $H_ω$ is bounded on the Hardy space $H^1$, and we show that for any $ω\in \widehat{\mathcal{D}}$ and $p\in (1,\infty)$, $H_ω:\,L^p_{[0,1)} \to H^p$ is bounded if and only if the Muckenhoupt type condition \begin{equation*} \sup\limits_{0<r<1}\left(1+\int_0^r \frac{1}{\widehatω(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 ω(t)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty, \end{equation*} holds. Moreover, we address the analogous question about the action of $H_ω$ on weighted Bergman spaces $A^p_ν$. |
| title | Hilbert-type operator induced by radial weight |
| topic | Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2007.15402 |