×

zbMATH — the first resource for mathematics

\({\bar \partial}\)-problem on weakly \(q\)-convex domains. (English) Zbl 0714.32006
We deal with a class of domains that we call weakly q-convex. These domains are generalizations of pseudoconvex domains. We show that we can solve the \({\bar \partial}\)-equation on weakly-\(q\)-convex domains on \((p,r)\)-forms for \(r\geq q\). We also show that we can prove other statements related to \({\bar \partial}\) in weakly \(q\)-convex domains that are true in pseudoconvex domains.
Reviewer: L.-H.Ho

MSC:
32F99 Geometric convexity in several complex variables
32T99 Pseudoconvex domains
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Catlin, D.: Necessary conditions for subellipticity of the \(\bar \partial \) -Neumann problem on pseudoconvex domains. Ann. Math. (2)117, 147-171 (1983) · Zbl 0552.32017 · doi:10.2307/2006974
[2] Catlin, D.: Subelliptic estimates for the \(\bar \partial \) -Neumann problem on pseudoconvex domains. Ann. Math. (2)126, 131-191 (1987) · Zbl 0627.32013 · doi:10.2307/1971347
[3] D’Angelo, J.: Real hypersurfaces, orders of contact, and applications. Ann. Math. (2)115, 615-637 (1982) · Zbl 0488.32008 · doi:10.2307/2007015
[4] Fischer, W., Lieb, I.: Lokale Kerne und beschr?nkte L?sungen f?r den \(\bar \partial \) -Opertaor aufq-konvexen Gebieten. Math. Ann.208, 249-265 (1974) · Zbl 0277.35045 · doi:10.1007/BF01419585
[5] Henkin, G.M., Leiterer, J.: Andreotti-Grauert theory by integral formulas. Basel: Birkh?user 1988 · Zbl 0654.32001
[6] Ho, L.: Subellipticity of the \(\bar \partial \) -Neumann problem on nonpseudoconvex domains. Trans. Am. Math. Soc.291, 43-73 (1985) · Zbl 0594.35074
[7] Ho, L.: Subellipticity estimate for the \(\bar \partial \) -Neumann problem forn?1 forms. Trans. Am. Math. Soc., to appear
[8] H?rmander, L.:L 2 estimates and existence theorems for the \(\bar \partial \) -operator. Acta Math.113, 89-152 (1965) · Zbl 0158.11002 · doi:10.1007/BF02391775
[9] H?rmander, L.: An introduction to complex analysis in several variables. Princeton: Van Nostrand 1966
[10] Kohn, J.J.: Global regularity for \(\bar \partial \) on weakly pseudoconvex manifolds. Trans. Am. Math. Soc.81, 273-292 (1973) · Zbl 0276.35071
[11] Kohn, J.J.: Subellipticity of the \(\bar \partial \) -Neumann problem on pseudoconvex domains: sufficient conditions. Acta Math.142, 79-122 (1979) · Zbl 0395.35069 · doi:10.1007/BF02395058
[12] Kohn, personal communication
[13] Post, S.: Finite type and subelliptic estimates for the \(\bar \partial \) -Neumann problem. Ph.D. thesis, Princeton University, Princeton, N.J., 1983
[14] Schmalz, G.: Solution of the \(\bar \partial \) -equation with uniform estimates on strictlyq-convex domains with nonsmooth boundary. Math. Z.202, 409-430 (1989) · Zbl 0662.32017 · doi:10.1007/BF01159970
[15] Shaw, M.: Global solvability and regularity for \(\bar \partial \) on an annulus between two weakly pseudoconvex domains. Trans. Am. Math. Soc.291, 255-267 (1985) · Zbl 0594.35010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.