Sobolev and Hölder estimates for homotopy operators of the $\overline\partial$-equation on convex domains of finite multitype
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910811287453696 |
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| author | Yao, Liding |
| author_facet | Yao, Liding |
| contents | We construct homotopy formulas for the $\overline\partial$-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain $Ω\subset\mathbb C^n$ that has $q$-type $m_q$ for $1\le q\le n$, our $\overline\partial$ solution operator $\mathcal H_q$ on $(0,q)$-forms has (fractional) Sobolev boundedness $\mathcal H_q:H^{s,p}\to H^{s+1/m_q,p}$ and Hölder-Zygmund boundedness $\mathcal H_q:\mathscr C^s\to\mathscr C^{s+1/m_q}$ for all $s\in\mathbb R$ and $1<p<\infty$. We also show the $L^p$-boundedness $\mathcal H_q:H^{s,p}\to H^{s,pr_q/(r_q-p)}$ for all $s\in\mathbb R$ and $1<p<r_q$, where $r_q:=(n-q+1)m_q+2q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_15830 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sobolev and Hölder estimates for homotopy operators of the $\overline\partial$-equation on convex domains of finite multitype Yao, Liding Complex Variables Analysis of PDEs 32A26 (primary) 32T25, 32W05 and 46E35 (secondary) We construct homotopy formulas for the $\overline\partial$-equation on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain $Ω\subset\mathbb C^n$ that has $q$-type $m_q$ for $1\le q\le n$, our $\overline\partial$ solution operator $\mathcal H_q$ on $(0,q)$-forms has (fractional) Sobolev boundedness $\mathcal H_q:H^{s,p}\to H^{s+1/m_q,p}$ and Hölder-Zygmund boundedness $\mathcal H_q:\mathscr C^s\to\mathscr C^{s+1/m_q}$ for all $s\in\mathbb R$ and $1<p<\infty$. We also show the $L^p$-boundedness $\mathcal H_q:H^{s,p}\to H^{s,pr_q/(r_q-p)}$ for all $s\in\mathbb R$ and $1<p<r_q$, where $r_q:=(n-q+1)m_q+2q$. |
| title | Sobolev and Hölder estimates for homotopy operators of the $\overline\partial$-equation on convex domains of finite multitype |
| topic | Complex Variables Analysis of PDEs 32A26 (primary) 32T25, 32W05 and 46E35 (secondary) |
| url | https://arxiv.org/abs/2210.15830 |