𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr; Osama Abdelkader

Czechoslovak Mathematical Journal (2017)

  • Issue: 2, page 515-523
  • ISSN: 0011-4642

Abstract

top
Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there exists a form u in 𝒞 0 , q - 1 k - ε ( X D , E ) with compact support such that ¯ u = f in X D ¯ . Applications are given for a separation theorem of Andreotti-Vesentini type in 𝒞 k -setting and for the solvability of the ¯ -equation for currents.

How to cite

top

Khidr, Shaban, and Abdelkader, Osama. "$\mathcal {C}^k$-regularity for the $\bar{\partial }$-equation with a support condition." Czechoslovak Mathematical Journal (2017): 515-523. <http://eudml.org/doc/288190>.

@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},
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
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 -

References

top
  1. Andreotti, A., Hill, C. D., E. E. Levi convexity and the Hans Lewy problem I: Reduction to vanishing theorems, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 26 (1972), 325-363. (1972) Zbl0256.32007MR0460725
  2. Andreotti, A., Hill, C. D., E. E. Levi convexity and the Hans Lewy problem II: Vanishing theorems, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 26 (1972), 747-806. (1972) Zbl0283.32013MR0477150
  3. Barkatou, M.-Y., Khidr, S., 10.1002/mana.200910063, Math. Nachr. 284 (2011), 2024-2031. (2011) Zbl1227.32016MR2844676DOI10.1002/mana.200910063
  4. Brinkschulte, J., 10.1007/BF02385479, Ark. Mat. 42 (2004), 259-282. (2004) Zbl1078.32023MR2101387DOI10.1007/BF02385479
  5. Grauert, H., Kantenkohomologie, Compos. Math. 44 (1981), 79-101 German. (1981) Zbl0512.32011MR0662457
  6. Henkin, G. M., Leiterer, J., 10.1007/978-1-4899-6724-4, Progress in Mathematics 74, Birkhäuser, Boston (1988). (1988) Zbl0654.32002MR0986248DOI10.1007/978-1-4899-6724-4
  7. Khidr, S., Barkatou, M.-Y., Global solutions with 𝒞 k -estimates for ¯ -equations on q -concave intersections, Electron. J. Differ. Equ. 2013 (2013), Paper No. 62, 10 pages. (2013) Zbl1287.32002MR3040639
  8. Laurent-Thiébaut, C., Leiterer, J., The Andreotti-Vesentini separation theorem with C k estimates and extension of CR-forms, Several Complex Variables, Proc. Mittag-Leffler Inst., Stockholm, 1987/1988 Math. Notes 38, Princeton Univ. Press, Princeton (1993), 416-439. (1993) Zbl0776.32012MR1207871
  9. Lieb, I., Range, R. M., 10.1007/BF01578911, Math. Ann. 253 (1980), 145-164 German. (1980) Zbl0441.32007MR0597825DOI10.1007/BF01578911
  10. Michel, J., 10.1007/BF01474180, Math. Ann. 280 (1988), 45-68 German. (1988) Zbl0617.32032MR0928297DOI10.1007/BF01474180
  11. Michel, J., Perotti, A., 10.1007/BF02570747, Math. Z. 203 (1990), 415-427. (1990) Zbl0673.32019MR1038709DOI10.1007/BF02570747
  12. Ricard, H., 10.1007/s00209-003-0504-4, Math. Z. 244 (2003), 349-398 French. (2003) Zbl1036.32012MR1992543DOI10.1007/s00209-003-0504-4
  13. Sambou, S., 10.1002/1522-2616(200202)235:1<179::AID-MANA179>3.0.CO;2-8, Math. Nachr. 235 (2002), 179-190 French. (2002) Zbl1007.32012MR1889284DOI10.1002/1522-2616(200202)235:1<179::AID-MANA179>3.0.CO;2-8
  14. Sambou, S., 10.5802/afst.1020, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 11 (2002), 105-129 French. (2002) Zbl1080.32502MR1986385DOI10.5802/afst.1020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.