How to Recognise Extension domains
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917437135388672 |
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| author | Mishra, Riddhi Mohanta, Kaushik |
| author_facet | Mishra, Riddhi Mohanta, Kaushik |
| contents | Let $Ω\subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We characterize $(1,p)$-extension domains in terms of inequalities of Bourgain--Brezis--Mironescu type. More precisely, we show that $Ω$ is a $(1,p)$-extension domain if and only if it is Ahlfors regular and satisfies, for all $f \in \dot{W}^{1,p}(Ω)$, \[(1-s)[f]_{W^{s,p}(Ω)}^p \leq C [f]_{W^{1,p}(Ω)}^p,\] for all $s$ sufficiently close to $1$, where $C > 0$ is a constant independent of $s$ and $f$.
As a key ingredient, we establish a fractional Poincaré-type inequality under the assumption of Ahlfors regularity alone, improving a result of Ponce (2004). As a further application, we prove that, under a mild Hausdorff measure condition on the boundary $\partial Ω$, fractional extension (from $\dot{W}^{1,p}(Ω)$ to $\dot{W}^{s,p}(\mathbb{R}^n)$) at a single exponent $s > 1/p$ self-improves to full first-order Sobolev extension (from $\dot{W}^{1,p}(Ω)$ to $\dot{W}^{1,p}(\mathbb{R}^n)$).
These results clarify the role of nonlocal estimates in the geometry of Sobolev extension domains. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23598 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | How to Recognise Extension domains Mishra, Riddhi Mohanta, Kaushik Functional Analysis Primary 46E35, Secondary 26D10, 28A75, 30L05 Let $Ω\subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We characterize $(1,p)$-extension domains in terms of inequalities of Bourgain--Brezis--Mironescu type. More precisely, we show that $Ω$ is a $(1,p)$-extension domain if and only if it is Ahlfors regular and satisfies, for all $f \in \dot{W}^{1,p}(Ω)$, \[(1-s)[f]_{W^{s,p}(Ω)}^p \leq C [f]_{W^{1,p}(Ω)}^p,\] for all $s$ sufficiently close to $1$, where $C > 0$ is a constant independent of $s$ and $f$. As a key ingredient, we establish a fractional Poincaré-type inequality under the assumption of Ahlfors regularity alone, improving a result of Ponce (2004). As a further application, we prove that, under a mild Hausdorff measure condition on the boundary $\partial Ω$, fractional extension (from $\dot{W}^{1,p}(Ω)$ to $\dot{W}^{s,p}(\mathbb{R}^n)$) at a single exponent $s > 1/p$ self-improves to full first-order Sobolev extension (from $\dot{W}^{1,p}(Ω)$ to $\dot{W}^{1,p}(\mathbb{R}^n)$). These results clarify the role of nonlocal estimates in the geometry of Sobolev extension domains. |
| title | How to Recognise Extension domains |
| topic | Functional Analysis Primary 46E35, Secondary 26D10, 28A75, 30L05 |
| url | https://arxiv.org/abs/2604.23598 |