Abstract
For a domain Ω of \({\mathbb{C}}^N \) we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the \(\bar \partial\)-complex for forms with \(C^\infty (\bar \Omega )\)-coefficients in degree \( \geqslant q + 1\). All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the L 2-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-L 2 and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.
Similar content being viewed by others
References
Andreotti, A. and Grauert, H.: Théorèmes definitude pour la cohomologie des éspaces complexes, Bull. Soc. Math. France 90 (1962), 193-259.
Dufresnoy, A.: Sur l'operateur \(\bar \partial \) et les fonctions différentiables au sens deWhitney, Ann. Inst. Fourier 29 (1) (1979), 229-238.
Grauert, H.: Kantenkohomologie, Compositio Math. 44 (1981), 79-101.
Henkin, G. M. and Leiterer, J.: Andreotti-Grauert Theory by Integral Formulas, Progr. in Math. 74, BirkhÌuser, Basel, 1988.
Hîrmander, L.: An Introduction to Complex Analysis in Several Complex Variables, Van Nostrand, Princeton, NJ 1966.
Hîrmander, L.: L 2-estimates and existence theorems for the \(\bar \partial \) operator, Acta Math. 113 (1965), 89-152.
Kohn, J. J.: Regularity at the boundary of the \(\bar \partial \)-Neumann problem, Proc. Nat. Acad. Sci. U.S.A 49 (1963), 206-213.
Michel, V.: Sur la régularité C ∞ du \(\bar \partial \) au bord d'un domaine de ℂn dont la forme de Levi á exactement valeurs propres strictement negatives, Math. Ann. 195 (1993), 131-165.
Treves, F.: Homotopy Formulas in the Tangential Cauchy-Riemann Complex, Mem. Amer. Math. Soc. Providence, Rhode Island, 1990.
Tumanov, A.: Extending CR functions on a manifold of finite type over a wedge, Mat. Sb. 136 (1988), 129-140.
Zampieri, G.: Simple sheaves along dihedral Lagrangians, J. Anal. Math. Jerusalem 66 (1995), 331-344.
Zampieri, G.: L 2-estimates with Levi-singular weight, and existence for \(\bar \partial \), J. Anal. Math. Jerusalem 74 (1998), 99-112.
Zampieri, G.: Soluability of \(\bar \partial \) with C ∞ regularity up to the boundary on wedges of ℂN, Israel J. Math. (2000).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zampieri, G. q-Pseudoconvexity and Regularity at the Boundary for Solutions of the \(\bar \partial \)-problem. Compositio Mathematica 121, 155–161 (2000). https://doi.org/10.1023/A:1001811318865
Issue Date:
DOI: https://doi.org/10.1023/A:1001811318865