${𝒞}^k$-regularity for the $\overline{\partial }$-equation with a support condition

@article{Khidr2017,
abstract = {Let $D$ be a $\mathcal \{C\}^d$$q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, 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 $\bar\{\partial \}$-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.},
author = {Khidr, Shaban, Abdelkader, Osama},
journal = {Czechoslovak Mathematical Journal},
keywords = {$\bar\{\partial \}$-equation; $q$-convexity; $\mathcal \{C\}^k$-estimate},
language = {eng},
number = {2},
pages = {515-523},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$\mathcal \{C\}^k$-regularity for the $\bar\{\partial \}$-equation with a support condition},
url = {http://eudml.org/doc/288190},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Khidr, Shaban
AU - Abdelkader, Osama
TI - $\mathcal {C}^k$-regularity for the $\bar{\partial }$-equation with a support condition
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 515
EP - 523
AB - Let $D$ be a $\mathcal {C}^d$$q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, 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 $\bar{\partial }$-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.
LA - eng
KW - $\bar{\partial }$-equation; $q$-convexity; $\mathcal {C}^k$-estimate
UR - http://eudml.org/doc/288190
ER -