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Lp-ESTIMATES FOR THE ${\bar{\partial}}$-EQUATION WITH EXACT SUPPORT ON q-CONVEX INTERSECTIONS

  • Khidr, Shaban (Mathematics Department Faculty of Science University of Jeddah)
  • Received : 2016.10.28
  • Accepted : 2017.08.04
  • Published : 2018.01.01

Abstract

We construct bounded linear integral operators that giving solutions to the ${\bar{\partial}}$-equation in $L^p$-spaces and with compact supports on a q-convex intersection ($q{\geq}1$) with ${\mathcal{C}}^3$ boundary in $K{\ddot{a}}hler$ manifolds, and we apply it to obtain a Hartogs-like extension theorems for ${\bar{\partial}}$-closed forms for some bidegree.

Keywords

References

  1. O. Abdelkader and S. Khidr, Solutions to ${\bar{\partial}}$-euations on stronly pseudo-convex domains with $L^{p}$-estimates, Electron. J. Differential Equations 2004 (2004), no. 73, 1-9.
  2. O. Abdelkader and S. Khidr, Solutions to ${\bar{\partial}}$-equations on strongly q-convex domains with $L^{p}$-estimates, Int. J. Geom. Methods Mod. Phys. 1 (2004), no. 6, 739-749. https://doi.org/10.1142/S0219887804000368
  3. E. Amar, An Andreotti-Grauert theorem with $L^{r}$ estimates, arXiv: 1203.0759v7, 2014.
  4. E. Amar and S. Mongodi, On $L^{r}$ hypoellipticity of solutions with compact support of the Cauchy-Riemann equation, Ann. Mat. Pura Appl. 193 (2014), no. 4, 999-1018. https://doi.org/10.1007/s10231-012-0312-8
  5. A. Andreotti and C. D. Hill, E. E. Levi convexity and the Hans Lewy problem I: Reduction to vanishing theorems, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), no. 2, 325-363.
  6. A. Andreotti and C. D. Hill, E. E. Levi convexity and the Hans Lewy problem II: Vanishing theorems, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), no. 4, 747-806.
  7. A. Andreotti and A. Kas, Duality on complex spaces, Ann. Scoula Norm. Sup. Pisa (3) 27 (1973), no. 2, 187-263.
  8. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2010.
  9. J. Brinkschulte, The ${\bar{\partial}}$-problem with support conditions on some weakly pseudoconvex domains, Ark. Mat. 42 (2004), no. 2, 259-282. https://doi.org/10.1007/BF02385479
  10. D. Chakrabarti and M.-C. Shaw, $L^{2}$ Serre duality on domains in complex manifolds and applications, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3529-3554. https://doi.org/10.1090/S0002-9947-2012-05511-5
  11. S.-C. Chen and M.-C. Shaw, Partial Differential Equations in Several Complex Variables, AMS/IP Stud. Adv. Math., 19, Amer. Math. Soc. Providence, R.I., 2001.
  12. M. Derridj, Le probleme de Cauchy pour ${\bar{\partial}}$ et application, Ann Sci. Ecole Norm. Sup. (4) 17 (1984), no. 3, 439-449. https://doi.org/10.24033/asens.1479
  13. M. Derridj, Regularite pour ${\bar{\partial}}$ dans quelques domaines faiblement pseudo-convexes, J. Differential Geom. 13 (1978), no. 4, 559-576. https://doi.org/10.4310/jdg/1214434708
  14. M. Derridj, Inegalites de Carleman et extension locale des fonctions holomorphes, Ann. Scuola Sup. Pisa Cl. Sci. (4) 9 (1982), no. 4, 645-669.
  15. G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, Springer-Verlag, New York, 2009.
  16. G. M. Henkin and J. Leiterer, Andreoutti-Grauert Theory by Integral Formulas, Progress in Math. 74, Birkhauser-Verlag, Boston, 1988.
  17. S. Khidr, Solving ${\bar{\partial}}$ with $L^{p}$-estimates on q-convex intersections in complex manifold, Complex Var. Elliptic Equ. 53 (2008), no. 3, 253-263. https://doi.org/10.1080/17476930701685783
  18. M. Landucci, Cauchy Problem for ${\bar{\partial}}$-operator in strictly pseudoconvex domains, Boll. Un. Mat. Ital. (5) 13A (1976), no. 1, 180-185.
  19. C. Laurent-Thiebaut, Theorie $L^{p}$ pour l'equation de Cauchy-Riemann, Ann. Fac. Sci. Toulouse Math. (6) 24 (2015), no. 2, 251-279. https://doi.org/10.5802/afst.1448
  20. L. Ma and S. K. Vassiliaduo, $L^{p}$-estimates for the Cauchy-Riemann operator on q-convex intersections in ${\mathbb{C}}n$, Manuscr. Math. 103 (2000), no. 4, 413-433. https://doi.org/10.1007/PL00005861
  21. C. Menini, Estimations pour la resolution du ${\bar{\partial}}$ sur une intersection d'Ouverts strictement pseudoconvexes, Math. Z. 225 (1997), no. 1, 87-93. https://doi.org/10.1007/PL00004305
  22. H. Ricard, Estimations $C^{k}$ pour l'Operator de Cauchy-Riemann sur des domaines a Coins q-Convexes et q-Concaves, Math. Z. 244 (2003), no. 2, 349-398. https://doi.org/10.1007/s00209-003-0504-4
  23. S. Sambou, Resolution du ${\bar{\partial}}$ pour les courants prolongeables, Math. Nachr. 235 (2002), no. 1, 179-190. https://doi.org/10.1002/1522-2616(200202)235:1<179::AID-MANA179>3.0.CO;2-8
  24. S. Sambou, Resolution du ${\bar{\partial}}$ pour les courants prolongeables definis dans un anneau, Ann. Fac. Sci. Toulouse Math. (6) 11 (2002), no. 1, 105-129. https://doi.org/10.5802/afst.1020