Abstract
Solvability for\(\bar \partial \) with regularity at the boundary of a domain Ω ⊂⊂ ℂn for forms of any degreek≥1 was characterized by pseudoconvexity of ϖΩ in [16]. It is proved here thatq-pseudoconvexity suffices to guarantee solvability of forms of degreek≥q+1. The method relies on theL 2 estimates in [13], and on their Sobolev version in [15] and [16].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Ahn, Global boundary regularity of the\(\bar \partial \) on q-pseudoconvex domains, preprint.
A. Andreotti and H. Grauert,Théorèmes de finitude pour la cohomologie des éspaces complexes, Bull. Soc. Math. France90 (1962), 193–259.
L. Baracco and G. Zampieri, Boundary regularity for\(\bar \partial \) on q-pseudoconvex wedges of ℂn, to appear.
D. Barrett,Behavior of the Bergman projection on the Diederich-Fornaess worm, Acta Math.168 (1992), 1–10.
S. Bell and E. Ligocka,A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math.57 (1980), 285–289.
M. Christ, Global C∞ irregularity of the\(\bar \partial \) problem for worm domains, J. Amer. Math. Soc.9 (1996), 1171–1185.
M. Derridj, On some estimates for the\(\bar \partial \)-Nuemann problem in nonpseudoconvex domains, Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978, Part 2, pp. 77–83.
M. Derridj, Inégalités à priori et estimations sous-elliptiques pour\(\bar \partial \) dans les ouverts non pseudoconvexes, Math. Ann.249 (1980), 27–48.
M. Derridj and D. Tartakoff, Sur la régulularité analytique globale des solutions du problème de Neumann pour\(\bar \partial \), Sém. Goulaouic-Schwartz (1976).
K. Diederich and J. E. Fornaess,Pseudoconvex domains: an example with non-trivial Nebenhülle, Math. Ann.225 (1977), 275–292.
A. Dufresnoy, Sur l'opérateur\(\bar \partial \) et les fonctions différentiables au sens de Whitney, Ann. Inst. Fourier (Grenoble)29 (1979), 229–238.
G. M. Henkin,H. Lewy's equation and analysis on pseudoconvex manifolds (Russian), I, Uspehi Mat. Nauk32 (1977), 57–118.
L. Hörmander, L2 estimates and existence theorems for the\(\bar \partial \) operator, Acta Math.113 (1965), 89–152.
L. Hörmander,An Introduction to Complex Analysis in Several Complex Variables, Van Nostrand., Princeton, N. J., 1966.
J. J. Kohn, Global regularity for\(\bar \partial \) ion weakly pseudo-convex manifolds, Trans. Amer. Math. Soc.181 (1973), 273–292.
J. J. Kohn,Methods of partial differential equations in complex analysis, Proc. Sympos. Pure Math.30 (1977), 215–237.
L. Rothschild and E. M. Stein,Hypoellitic differential operators and nilpotent groups, Acta Math.137 (1976), 257–320.
G. Zampieri, q-pseudoconvexity and regularity at the boundary for solutions of the\(\bar \partial \), Compositio Math.121 (2000), 155–162.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baracco, L., Zampieri, G. Regularity at the boundary for v onQ-pseudoconvex domainsonQ-pseudoconvex domains. J. Anal. Math. 95, 45–61 (2005). https://doi.org/10.1007/BF02791496
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02791496