Hilbert matrix operator on bound analytic functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913562469859328 |
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| author | Guo, Yuting Tang, Pengcheng |
| author_facet | Guo, Yuting Tang, Pengcheng |
| contents | It is well known that the Hilbert matrix operator $\mathcal {H}$ is bounded from $H^{\infty}$ to the mean Lipschitz spaces $Λ^{p}_{\frac{1}{p}}$ for all $1<p<\infty$. In this paper, we prove that the range of Hilbert matrix operator $\mathcal {H}$ acting on $H^{\infty}$ is contained in certain Zygmund-type space (denoted by $Λ^{1.*}_{1}$), which is strictly smaller than $\cap_{p>1}Λ^{p}_{\frac{1}{p}}$. We also provide explicit upper and lower bounds for the norm of the Hilbert matrix $\mathcal {H}$ acting from $H^{\infty}$ to $Λ^{1.*}_{1}$. Additionally, we also characterize the positive Borel measures $μ$ such that the generalized Hilbert matrix operator $\mathcal {H}_μ$ is bounded from $H^{\infty}$ to the Hardy space $H^{q}$. This part is a continuation of the work of Chatzifountas, Girela and Peláez [J. Math. Anal. Appl. 413 (2014) 154--168] regarding $\mathcal {H}_μ$ on Hardy spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18682 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbert matrix operator on bound analytic functions Guo, Yuting Tang, Pengcheng Functional Analysis Complex Variables It is well known that the Hilbert matrix operator $\mathcal {H}$ is bounded from $H^{\infty}$ to the mean Lipschitz spaces $Λ^{p}_{\frac{1}{p}}$ for all $1<p<\infty$. In this paper, we prove that the range of Hilbert matrix operator $\mathcal {H}$ acting on $H^{\infty}$ is contained in certain Zygmund-type space (denoted by $Λ^{1.*}_{1}$), which is strictly smaller than $\cap_{p>1}Λ^{p}_{\frac{1}{p}}$. We also provide explicit upper and lower bounds for the norm of the Hilbert matrix $\mathcal {H}$ acting from $H^{\infty}$ to $Λ^{1.*}_{1}$. Additionally, we also characterize the positive Borel measures $μ$ such that the generalized Hilbert matrix operator $\mathcal {H}_μ$ is bounded from $H^{\infty}$ to the Hardy space $H^{q}$. This part is a continuation of the work of Chatzifountas, Girela and Peláez [J. Math. Anal. Appl. 413 (2014) 154--168] regarding $\mathcal {H}_μ$ on Hardy spaces. |
| title | Hilbert matrix operator on bound analytic functions |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2410.18682 |