| Title:
|
$\mathcal C^k$-regularity for the $\bar \partial $-equation with a support condition (English) |
| Author:
|
Khidr, Shaban |
| Author:
|
Abdelkader, Osama |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
515-523 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $D$ be a $\mathcal {C}^d$ $q$-convex intersection, $d\geq 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, $\mathcal C^k$-estimates, $k=2, 3, \dots , \infty $, for solutions to the \hbox {$\bar \partial $-equation} with small loss of smoothness are obtained for $E$-valued $(0, s)$-forms on $D$ when $ n-q\le s\le n$. In addition, we solve the $\bar \partial $-equation with a support condition in $\mathcal C^k$-spaces. More precisely, we prove that for a $\bar \partial $-closed form $f$ in $\mathcal C_{0,q}^{k}(X\setminus D, E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\varepsilon $ with $0<\varepsilon <1$ there exists a form $u$ in $\mathcal C_{0,q-1}^{k-\varepsilon }(X\setminus D, E)$ with compact support such that $\bar {\partial }u=f$ in $X\setminus \overline D$. Applications are given for a separation theorem of Andreotti-Vesentini type in $\mathcal C^k$-setting and for the solvability of the $\bar \partial $-equation for currents. (English) |
| Keyword:
|
$\bar \partial $-equation |
| Keyword:
|
$q$-convexity |
| Keyword:
|
$\mathcal C^k$-estimate |
| MSC:
|
32F10 |
| MSC:
|
32W05 |
| idZBL:
|
Zbl 06738534 |
| idMR:
|
MR3661056 |
| DOI:
|
10.21136/CMJ.2017.0039-16 |
| . |
| Date available:
|
2017-06-01T14:31:30Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146771 |
| . |
| Reference:
|
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| Reference:
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