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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2605.13787 |
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| _version_ | 1866910217296412672 |
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| author | Idrissi, H. Bahajji-El El-Fallah, O. Elmadani, Y. Hanine, A. |
| author_facet | Idrissi, H. Bahajji-El El-Fallah, O. Elmadani, Y. Hanine, A. |
| contents | In this paper, we consider weighted Dirichlet spaces $\cD_ω$, where $ω$ is a positive superharmonic weight on the unit disc $\DD$. These spaces include the standard weighted Dirichlet spaces $\cD_α$ and appear in the description of their invariant subspaces. Our goal is to study the spaces $\cD_ω$. We show that an explicit description of invariant subspaces reduces to the description of those generated by a bounded outer function, and then to the problem of describing cyclic functions, known as the Brown--Shields conjecture. We develop tools, analogous to those used in the harmonic case, that are needed to treat this problem for superharmonically weighted Dirichlet spaces $\cD_ω$. In particular, we obtain a formula for the Dirichlet integral of outer functions of Carleson--Richter--Sundberg type, estimates for the norm of the reproducing kernel of $\cD_ω$, and several properties on the capacity associated with $\cD_ω$. Using these tools, we provide a description of invariant subspaces when the measure $Δω$ is finite measure or if the $\supp(Δω)\cap \TT$ is countable, where $\TT$ denotes the unit circle. Finally, we prove that a smooth outer function $f \in \cD_α$ such that $\cZ (f) $ is "regular" is cyclic in $\cD_α$ if and only if $c_{α}(\cZ(f))= 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13787 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Superharmonically Weighted Dirichlet Spaces Idrissi, H. Bahajji-El El-Fallah, O. Elmadani, Y. Hanine, A. Functional Analysis Classical Analysis and ODEs 46E22, 47B32, 47A15, 31A15, 31A20 In this paper, we consider weighted Dirichlet spaces $\cD_ω$, where $ω$ is a positive superharmonic weight on the unit disc $\DD$. These spaces include the standard weighted Dirichlet spaces $\cD_α$ and appear in the description of their invariant subspaces. Our goal is to study the spaces $\cD_ω$. We show that an explicit description of invariant subspaces reduces to the description of those generated by a bounded outer function, and then to the problem of describing cyclic functions, known as the Brown--Shields conjecture. We develop tools, analogous to those used in the harmonic case, that are needed to treat this problem for superharmonically weighted Dirichlet spaces $\cD_ω$. In particular, we obtain a formula for the Dirichlet integral of outer functions of Carleson--Richter--Sundberg type, estimates for the norm of the reproducing kernel of $\cD_ω$, and several properties on the capacity associated with $\cD_ω$. Using these tools, we provide a description of invariant subspaces when the measure $Δω$ is finite measure or if the $\supp(Δω)\cap \TT$ is countable, where $\TT$ denotes the unit circle. Finally, we prove that a smooth outer function $f \in \cD_α$ such that $\cZ (f) $ is "regular" is cyclic in $\cD_α$ if and only if $c_{α}(\cZ(f))= 0$. |
| title | Superharmonically Weighted Dirichlet Spaces |
| topic | Functional Analysis Classical Analysis and ODEs 46E22, 47B32, 47A15, 31A15, 31A20 |
| url | https://arxiv.org/abs/2605.13787 |