Bergman projection induced by radial weight acting on growth spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914849365164032 |
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| author | Moreno, Álvaro Miguel Peláez, José Ángel Taskinen, Jari |
| author_facet | Moreno, Álvaro Miguel Peláez, José Ángel Taskinen, Jari |
| contents | Let $ω$ be a radial weight on the unit disc of the complex plane $\mathbb{D}$ and denote $ω_x =\int_0^1 s^x ω(s)\,ds$, $x\ge 0$, for the moments of $ω$ and $\widehatω(r)=\int_r^1 ω(s)\,ds$ for the tail integrals. A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if satisfies the upper doubling condition $$\sup_{0<r<1}\frac{\widehatω(r)}{\widehatω\left(\frac{1+r}{2}\right)}<\infty.$$
If $ν$ or $ω$ belongs to $\widehat{\mathcal{D}}$, it is described the boundedness of the Bergman projection $P_ω$ induced by $ω$ on the growth space $L^\infty_{\widehatν} =\{ f: \|f\|_{\infty,v}={ esssup}_{z\in\mathbb{D}} |f(z)|\widehatν(z)<\infty\}$ in terms of neat conditions on the moments and/or the tail integrals of $ω$ and $ν$. Moreover, it is solved the analogous problem for $P_ω$ from $L^\infty_{\widehatν}$ to the Bloch type space $B^\infty_{\widehatν}$ of analytic functions such that $\sup_{z\in \mathbb{D}}(1-|z|)\widehatν(z) |f'(z)|<\infty.$ We also study similar questions for exponentially decreasing radial weights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bergman projection induced by radial weight acting on growth spaces Moreno, Álvaro Miguel Peláez, José Ángel Taskinen, Jari Complex Variables Let $ω$ be a radial weight on the unit disc of the complex plane $\mathbb{D}$ and denote $ω_x =\int_0^1 s^x ω(s)\,ds$, $x\ge 0$, for the moments of $ω$ and $\widehatω(r)=\int_r^1 ω(s)\,ds$ for the tail integrals. A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if satisfies the upper doubling condition $$\sup_{0<r<1}\frac{\widehatω(r)}{\widehatω\left(\frac{1+r}{2}\right)}<\infty.$$ If $ν$ or $ω$ belongs to $\widehat{\mathcal{D}}$, it is described the boundedness of the Bergman projection $P_ω$ induced by $ω$ on the growth space $L^\infty_{\widehatν} =\{ f: \|f\|_{\infty,v}={ esssup}_{z\in\mathbb{D}} |f(z)|\widehatν(z)<\infty\}$ in terms of neat conditions on the moments and/or the tail integrals of $ω$ and $ν$. Moreover, it is solved the analogous problem for $P_ω$ from $L^\infty_{\widehatν}$ to the Bloch type space $B^\infty_{\widehatν}$ of analytic functions such that $\sup_{z\in \mathbb{D}}(1-|z|)\widehatν(z) |f'(z)|<\infty.$ We also study similar questions for exponentially decreasing radial weights. |
| title | Bergman projection induced by radial weight acting on growth spaces |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2406.18446 |