Contractive Hardy--Littlewood inequalities in the Dirichlet range
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908596992737280 |
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| author | Brevig, Ole Fredrik Kulikov, Aleksei Seip, Kristian Zlotnikov, Ilya |
| author_facet | Brevig, Ole Fredrik Kulikov, Aleksei Seip, Kristian Zlotnikov, Ilya |
| contents | The class $A_α^p$ consists of those analytic functions $f$ in the unit disc such that \[\|f\|_{α,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{α-1} \,dr < \infty,\] where $M_p^p(r,f)$ is the radial integral mean of $|f|^p$ and $0<α, p <\infty$. For $α>1$, $A_α^p$ is the standard weighted Bergman space, and $A_1^p=H^p$. We consider $A_α^p$ for $0<α<1$ and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between $A_α^p$ and the classical Besov spaces. Our main result is the contractive inequality $\|f\|_{β,q} \leq \|f\|_{α,p}$, valid when $0<α<β<\infty$ and $α/p=β/q$. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author ($1\leq α<β$) and Llinares ($β=1$ and $p=2$). The extension of results from the classical range $1\leq α< \infty$ to the Dirichlet range $0<α<1$ uses arguments relying on analytic continuation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Contractive Hardy--Littlewood inequalities in the Dirichlet range Brevig, Ole Fredrik Kulikov, Aleksei Seip, Kristian Zlotnikov, Ilya Complex Variables Classical Analysis and ODEs Functional Analysis The class $A_α^p$ consists of those analytic functions $f$ in the unit disc such that \[\|f\|_{α,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{α-1} \,dr < \infty,\] where $M_p^p(r,f)$ is the radial integral mean of $|f|^p$ and $0<α, p <\infty$. For $α>1$, $A_α^p$ is the standard weighted Bergman space, and $A_1^p=H^p$. We consider $A_α^p$ for $0<α<1$ and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between $A_α^p$ and the classical Besov spaces. Our main result is the contractive inequality $\|f\|_{β,q} \leq \|f\|_{α,p}$, valid when $0<α<β<\infty$ and $α/p=β/q$. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author ($1\leq α<β$) and Llinares ($β=1$ and $p=2$). The extension of results from the classical range $1\leq α< \infty$ to the Dirichlet range $0<α<1$ uses arguments relying on analytic continuation. |
| title | Contractive Hardy--Littlewood inequalities in the Dirichlet range |
| topic | Complex Variables Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2510.14333 |