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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.08294 |
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| _version_ | 1866909250681307136 |
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| author | Charpentier, Stéphane Espoullier, Nicolas Zarouf, Rachid |
| author_facet | Charpentier, Stéphane Espoullier, Nicolas Zarouf, Rachid |
| contents | We prove the existence of functions $f$ in the Bloch space of the unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ with the property that, given any measurable function $φ$ on the unit sphere $\mathbb{S}_N$, there exists a sequence $(r_n)_n$, $r_n\in (0,1)$, converging to $1$, such that for every $w\in \mathbb{B}_N$, $$f(r_n(ζ-w)+w) \to φ(ζ)\text{ as }n\to \infty\text{, for almost every }ζ\in \mathbb{S}_N.$$ The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08294 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bloch functions with wild boundary behaviour in $\mathbb{C}^N$ Charpentier, Stéphane Espoullier, Nicolas Zarouf, Rachid Complex Variables We prove the existence of functions $f$ in the Bloch space of the unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ with the property that, given any measurable function $φ$ on the unit sphere $\mathbb{S}_N$, there exists a sequence $(r_n)_n$, $r_n\in (0,1)$, converging to $1$, such that for every $w\in \mathbb{B}_N$, $$f(r_n(ζ-w)+w) \to φ(ζ)\text{ as }n\to \infty\text{, for almost every }ζ\in \mathbb{S}_N.$$ The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc. |
| title | Bloch functions with wild boundary behaviour in $\mathbb{C}^N$ |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2407.08294 |