Tangential boundary behavior in Hilbert spaces of analytic functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911355343208448 |
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| author | Luo, Shuaibing Malman, Bartosz |
| author_facet | Luo, Shuaibing Malman, Bartosz |
| contents | Sarason's Hilbert space version of Carathéodory-Julia Theorem connects the non-tangential boundary behavior of functions in de Branges-Rovnyak space $H(b)$ with the existence of angular derivatives in the sense of Carathéodory for $b$, an analytic self-mapping of the unit disk. In this article, we continue the study of higher order extensions of this result that deal with derivatives of functions in $H(b)$, and we consider notions of approach regions more general than the non-tangential ones. Our main result generalizes the recent work of Duan-Li-Mashreghi on boundary behavior in model spaces to $H(b)$-spaces and to higher order derivatives, and we give a new self-contained proof of that result. It also generalizes earlier radial results of Fricain-Mashreghi. In relation to existence of angular derivatives, we show that in the classical Carathéodory-Julia Theorem one cannot replace the non-tangential approach region by any essentially larger region. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02194 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tangential boundary behavior in Hilbert spaces of analytic functions Luo, Shuaibing Malman, Bartosz Functional Analysis Complex Variables Sarason's Hilbert space version of Carathéodory-Julia Theorem connects the non-tangential boundary behavior of functions in de Branges-Rovnyak space $H(b)$ with the existence of angular derivatives in the sense of Carathéodory for $b$, an analytic self-mapping of the unit disk. In this article, we continue the study of higher order extensions of this result that deal with derivatives of functions in $H(b)$, and we consider notions of approach regions more general than the non-tangential ones. Our main result generalizes the recent work of Duan-Li-Mashreghi on boundary behavior in model spaces to $H(b)$-spaces and to higher order derivatives, and we give a new self-contained proof of that result. It also generalizes earlier radial results of Fricain-Mashreghi. In relation to existence of angular derivatives, we show that in the classical Carathéodory-Julia Theorem one cannot replace the non-tangential approach region by any essentially larger region. |
| title | Tangential boundary behavior in Hilbert spaces of analytic functions |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2601.02194 |