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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.04626 |
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| _version_ | 1866910105744703488 |
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| author | Boudabra, Maher |
| author_facet | Boudabra, Maher |
| contents | We introduce the space $\mathcal{W}^{s,p}(\mathbb{D})$ of analytic functions $u$ on the unit disc such that the radial restrictions $u_{r}(ξ):=u(rξ)$ satisfy the Gagliardo seminorm-only bound \[ \sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}<\infty, \] with no $\emph{a priori}$ control of $\sup_{r}\|u_{r}\|_{L^{p}(\mathbb{S}^{1})}$. Our main result shows that this assumption already forces $u\in H^{p}(\mathbb{D})$ and that the radial boundary trace $u^{*}$ belongs to $W^{s,p}(\mathbb{S}^{1})$, with $u_{r}\to u^{*}$ in $W^{s,p}(\mathbb{S}^{1})$ as $r\to1^{-}$. The key mechanism combines the mean-value property (which pins the constant mode at $u(0)$) with a fractional Poincar$é$ inequality on $\mathbb{S}^{1}$, recovering $L^{p}$ control from oscillation alone. As a consequence, the trace map $u\mapsto u^{*}$ is a surjective isomorphism $\mathcal{W}^{s,p}(\mathbb{D})\xrightarrow{\sim}B^{s}_{p,p,+}(\mathbb{S}^{1})$ with explicit norm equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04626 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A seminorm-only characterization of analytic Besov spaces on the disc Boudabra, Maher Analysis of PDEs We introduce the space $\mathcal{W}^{s,p}(\mathbb{D})$ of analytic functions $u$ on the unit disc such that the radial restrictions $u_{r}(ξ):=u(rξ)$ satisfy the Gagliardo seminorm-only bound \[ \sup_{0<r<1}[u_{r}]_{W^{s,p}(\mathbb{S}^{1})}<\infty, \] with no $\emph{a priori}$ control of $\sup_{r}\|u_{r}\|_{L^{p}(\mathbb{S}^{1})}$. Our main result shows that this assumption already forces $u\in H^{p}(\mathbb{D})$ and that the radial boundary trace $u^{*}$ belongs to $W^{s,p}(\mathbb{S}^{1})$, with $u_{r}\to u^{*}$ in $W^{s,p}(\mathbb{S}^{1})$ as $r\to1^{-}$. The key mechanism combines the mean-value property (which pins the constant mode at $u(0)$) with a fractional Poincar$é$ inequality on $\mathbb{S}^{1}$, recovering $L^{p}$ control from oscillation alone. As a consequence, the trace map $u\mapsto u^{*}$ is a surjective isomorphism $\mathcal{W}^{s,p}(\mathbb{D})\xrightarrow{\sim}B^{s}_{p,p,+}(\mathbb{S}^{1})$ with explicit norm equivalence. |
| title | A seminorm-only characterization of analytic Besov spaces on the disc |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.04626 |