How smooth are restrictions of Besov functions?
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912649533456384 |
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| author | Brasseur, Julien |
| author_facet | Brasseur, Julien |
| contents | In a previous work, we showed that Besov spaces do not enjoy the restriction property unless $q\leq p$. Specifically, we proved that if $p<q$, then it is always possible to construct a function $f\in B_{p,q}^s(\mathbb{R}^N)$ such that $f(\cdot,y)\notin B_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$, while this "pathology" does not happen if $q\leq p$. We showed that the partial maps belong, in fact, to the Besov space of generalised smoothness $B_{p,q}^{(s,Ψ)}(\mathbb{R}^d)$ provided the function $Ψ$ satisfies a simple summability condition involving $p$ and $q$. This short note completes the picture by showing that this characterisation is sharp. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How smooth are restrictions of Besov functions? Brasseur, Julien Functional Analysis Analysis of PDEs Classical Analysis and ODEs In a previous work, we showed that Besov spaces do not enjoy the restriction property unless $q\leq p$. Specifically, we proved that if $p<q$, then it is always possible to construct a function $f\in B_{p,q}^s(\mathbb{R}^N)$ such that $f(\cdot,y)\notin B_{p,q}^s(\mathbb{R}^d)$ for a.e. $y\in \mathbb{R}^{N-d}$, while this "pathology" does not happen if $q\leq p$. We showed that the partial maps belong, in fact, to the Besov space of generalised smoothness $B_{p,q}^{(s,Ψ)}(\mathbb{R}^d)$ provided the function $Ψ$ satisfies a simple summability condition involving $p$ and $q$. This short note completes the picture by showing that this characterisation is sharp. |
| title | How smooth are restrictions of Besov functions? |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2509.07420 |