On $\overline\partial$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates
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| Format: | Preprint |
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2025
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| _version_ | 1866916594844696576 |
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| author | Yao, Liding Zhang, Yuan |
| author_facet | Yao, Liding Zhang, Yuan |
| contents | We construct homotopy formulae $f=\overline\partial\mathcal H_qf+\mathcal H_{q+1}\overline\partial f$ for $(0,q)$ forms on the product domain $Ω_1\times\dots\timesΩ_m$, where each $Ω_j$ is either a bounded Lipschitz domain in $\mathbb C^1$, a bounded strongly pseudoconvex domain with $C^2$ boundary, or a smooth convex domain of finite type. Such homotopy operators $\mathcal H_q$ yield solutions to the $\overline\partial$ equation with optimal Sobolev regularity $W^{k,p}\to W^{k,p}$ simultaneously for all $k\in\mathbb Z$ and $1<p<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $\overline\partial$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates Yao, Liding Zhang, Yuan Complex Variables 32A26 (primary) 32W05 and 46E35 (secondary) We construct homotopy formulae $f=\overline\partial\mathcal H_qf+\mathcal H_{q+1}\overline\partial f$ for $(0,q)$ forms on the product domain $Ω_1\times\dots\timesΩ_m$, where each $Ω_j$ is either a bounded Lipschitz domain in $\mathbb C^1$, a bounded strongly pseudoconvex domain with $C^2$ boundary, or a smooth convex domain of finite type. Such homotopy operators $\mathcal H_q$ yield solutions to the $\overline\partial$ equation with optimal Sobolev regularity $W^{k,p}\to W^{k,p}$ simultaneously for all $k\in\mathbb Z$ and $1<p<\infty$. |
| title | On $\overline\partial$ homotopy formulae for product domains: Nijenhuis-Woolf's formulae and optimal Sobolev estimates |
| topic | Complex Variables 32A26 (primary) 32W05 and 46E35 (secondary) |
| url | https://arxiv.org/abs/2502.00925 |