Lower bounds for the weak-type constants of the operators $Λ_m$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912367005138944 |
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| author | Strzelecki, Michał |
| author_facet | Strzelecki, Michał |
| contents | The operators $Λ_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $Λ_0$ is equal to $1/\ln(2)\approx 1.44$. We construct examples showing that the weak-type $(1,1)$ constant of $Λ_1$ is larger than $1.38$ and that the weak-type $(1,1)$ constant of $Λ_m$ does not tend to $1$ when $m\to\infty$. This disproves a conjecture of Gill [Mich. Math. J. 59 (2010), No. 2, 353-363]. We also prove a companion result for the adjoint operators. This is the arXiv version of the paper - it includes some additional discussion in the appendices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09340 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower bounds for the weak-type constants of the operators $Λ_m$ Strzelecki, Michał Classical Analysis and ODEs Functional Analysis 26D10 (Primary), 42B20 (Secondary) The operators $Λ_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $Λ_0$ is equal to $1/\ln(2)\approx 1.44$. We construct examples showing that the weak-type $(1,1)$ constant of $Λ_1$ is larger than $1.38$ and that the weak-type $(1,1)$ constant of $Λ_m$ does not tend to $1$ when $m\to\infty$. This disproves a conjecture of Gill [Mich. Math. J. 59 (2010), No. 2, 353-363]. We also prove a companion result for the adjoint operators. This is the arXiv version of the paper - it includes some additional discussion in the appendices. |
| title | Lower bounds for the weak-type constants of the operators $Λ_m$ |
| topic | Classical Analysis and ODEs Functional Analysis 26D10 (Primary), 42B20 (Secondary) |
| url | https://arxiv.org/abs/2411.09340 |