A universal $\overline\partial$ solution operator on nonsmooth strongly pseudoconvex domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915084024938496 |
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| author | Yao, Liding |
| author_facet | Yao, Liding |
| contents | We construct homotopy formulae $f=\overline\partial \mathcal H_q f+\mathcal H_{q+1}\overline\partial f$ on a bounded domain which is either $C^2$ strongly pseudoconvex or $C^{1,1}$ strongly $\mathbb C$-linearly convex. Such operators exhibit Sobolev estimates $\mathcal H_q:H^{s,p}\to H^{s+1/2,p}$ and Hölder-Zygmund estimates $\mathcal H_q:\mathscr C^s\to\mathscr C^{s+1/2}$ simultaneously for all $s\in\mathbb R$ and $1<p<\infty$. In particular this provides the existence and $\frac12$ estimate for solution operator on Sobolev space of negative index these domains. The construction uses a new decomposition for the commutator $[\overline\partial,\mathcal E]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20312 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A universal $\overline\partial$ solution operator on nonsmooth strongly pseudoconvex domains Yao, Liding Complex Variables 32A26 (primary) 32T25, 32W05 and 46E35 (secondary) We construct homotopy formulae $f=\overline\partial \mathcal H_q f+\mathcal H_{q+1}\overline\partial f$ on a bounded domain which is either $C^2$ strongly pseudoconvex or $C^{1,1}$ strongly $\mathbb C$-linearly convex. Such operators exhibit Sobolev estimates $\mathcal H_q:H^{s,p}\to H^{s+1/2,p}$ and Hölder-Zygmund estimates $\mathcal H_q:\mathscr C^s\to\mathscr C^{s+1/2}$ simultaneously for all $s\in\mathbb R$ and $1<p<\infty$. In particular this provides the existence and $\frac12$ estimate for solution operator on Sobolev space of negative index these domains. The construction uses a new decomposition for the commutator $[\overline\partial,\mathcal E]$. |
| title | A universal $\overline\partial$ solution operator on nonsmooth strongly pseudoconvex domains |
| topic | Complex Variables 32A26 (primary) 32T25, 32W05 and 46E35 (secondary) |
| url | https://arxiv.org/abs/2412.20312 |