Boundary behavior of functions in the Schur-Agler class of the polydisc
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913997765214208 |
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| author | Agler, Jim Evans, Connor Lykova, Zinaida Young, N. J. |
| author_facet | Agler, Jim Evans, Connor Lykova, Zinaida Young, N. J. |
| contents | We describe a generalization of the notion of a Hilbert space model of a function $φ$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $φ$ at a mild singularity $τ$ on the $d$-torus. We prove the existence of a generalized model with an enhanced continuity property at such a singularity $τ$. We use this result to prove the directional differentiability of a function $φ$ in the Schur-Agler class at a singular point on the $d$-torus for which the Carathéodory condition holds and to calculate the corresponding directional derivative. The results of this paper extend to the polydisc $\mathbb{D}^d$ results of Agler, McCarthy, Tully-Doyle and Young which generalized to the bidisc the classical Julia-Wolff-Carathéodory theorem about analytic self-maps of $\mathbb{D}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13742 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary behavior of functions in the Schur-Agler class of the polydisc Agler, Jim Evans, Connor Lykova, Zinaida Young, N. J. Complex Variables 32A30, 32S05, 47A56, 47A57 We describe a generalization of the notion of a Hilbert space model of a function $φ$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $φ$ at a mild singularity $τ$ on the $d$-torus. We prove the existence of a generalized model with an enhanced continuity property at such a singularity $τ$. We use this result to prove the directional differentiability of a function $φ$ in the Schur-Agler class at a singular point on the $d$-torus for which the Carathéodory condition holds and to calculate the corresponding directional derivative. The results of this paper extend to the polydisc $\mathbb{D}^d$ results of Agler, McCarthy, Tully-Doyle and Young which generalized to the bidisc the classical Julia-Wolff-Carathéodory theorem about analytic self-maps of $\mathbb{D}$. |
| title | Boundary behavior of functions in the Schur-Agler class of the polydisc |
| topic | Complex Variables 32A30, 32S05, 47A56, 47A57 |
| url | https://arxiv.org/abs/2508.13742 |