$A^p_α$ classes in the Dirichlet range: inner-outer factorization, Carleson measures and weak products
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arXiv
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| Natura: | Preprint |
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2026
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| author | Dayan, Alberto Llinares, Adrián Monsalve-López, Miguel |
| author_facet | Dayan, Alberto Llinares, Adrián Monsalve-López, Miguel |
| contents | We study properties of $A^p_α$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that
\[
\int_{\mathbb{D}} |f(z)|^{p-2} |f'(z)|^2 (1 - |z|^2)^α \, dA(z) < \infty,
\]
when $0<α< 1$ and $p > 0$. We answer in the negative two questions posed by Brevig et al. by showing that, if $p\ne2$ and $p > \frac{1}{2}$, $A^p_α$ is not a vector space and that the norm is in general not increasing in $p$. This is achieved by means of an equivalent description for $A^p_α$ which is given in terms of the Poisson integral of the boundary function of its inhabitants. Such norm also leads to a description of $A^p_α$ functions in the Dirichlet range given in terms of their inner and outer factors. As a corollary, we show that $A^1_α$ is contained in the weak product of a Dirichlet-type space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21347 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $A^p_α$ classes in the Dirichlet range: inner-outer factorization, Carleson measures and weak products Dayan, Alberto Llinares, Adrián Monsalve-López, Miguel Complex Variables Classical Analysis and ODEs Functional Analysis 47B91 (Primary), 30J05, 30H25 (Secondary) We study properties of $A^p_α$ spaces in the Dirichlet range, recently defined by Brevig, Kulikov, Seip and Zlotnikov as the set of all holomorphic functions on the unit disc $\mathbb{D}$ such that \[ \int_{\mathbb{D}} |f(z)|^{p-2} |f'(z)|^2 (1 - |z|^2)^α \, dA(z) < \infty, \] when $0<α< 1$ and $p > 0$. We answer in the negative two questions posed by Brevig et al. by showing that, if $p\ne2$ and $p > \frac{1}{2}$, $A^p_α$ is not a vector space and that the norm is in general not increasing in $p$. This is achieved by means of an equivalent description for $A^p_α$ which is given in terms of the Poisson integral of the boundary function of its inhabitants. Such norm also leads to a description of $A^p_α$ functions in the Dirichlet range given in terms of their inner and outer factors. As a corollary, we show that $A^1_α$ is contained in the weak product of a Dirichlet-type space. |
| title | $A^p_α$ classes in the Dirichlet range: inner-outer factorization, Carleson measures and weak products |
| topic | Complex Variables Classical Analysis and ODEs Functional Analysis 47B91 (Primary), 30J05, 30H25 (Secondary) |
| url | https://arxiv.org/abs/2604.21347 |