×

\(\mathcal C^{k+\alpha}\)-estimates for the \(\bar \partial\)-equation on the Hartogs triangle. (English) Zbl 0756.32002

On Hartogs triangle, \({\mathcal C}^{k+\alpha}\)-estimates for \(\overline\partial\) are shown for all non-negative \(k+\alpha\). This is done by pulling the problem back to the bidisc. A special solution, constructed there, can be pushed forward to the Hartogs triangle, in the course of which all estimates and regularity conditions are preserved.
Reviewer: L.Ma

MSC:

32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A40 Boundary behavior of holomorphic functions of several complex variables
32T99 Pseudoconvex domains
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Bertrams, J.: Das \(\bar \partial \) -Problem auf pseudokonvexen Polyedern nach Sergeev und Henkin. Bonn. Math. Schr.167 (1985) · Zbl 0597.32017
[2] Bertrams, J.: Randregularität von Lösungen der \(\bar \partial \) -Gleichung auf dem Polyzylinder und zweidimensionalen analytischen Polyedern. Bonn. Math. Schr.176 (1986) · Zbl 0622.32001
[3] Chaumat, J., Chollet, A.-M.: Noyaux pour résoudre l’équation \(\bar \partial \) dans des classes ultradifférentiables sur des compacts irréguliers de C n . In: Several complex variables. Proc. Mittag-Leffler Inst. 1987/1988. (Math. Notes, Princeton, Vol. 38) Princeton: Princeton University Press (to appear)
[4] Chaumat, J., Chollet, A.-M.: Régularité holderienne de l’operateur \(\bar \partial \) sur le triangle de Hartogs. (Preprint) · Zbl 0735.32004
[5] Hakim, M., Sibony, N.: Spectre de \(A(\bar \Omega )\) pour des domaines bornés faiblement pseudoconvexes réguliers. J. Funct. Anal.37 (no. 2), 127–135 (1980) · Zbl 0441.46044
[6] Kohn, J.J.: Global regularity for \(\bar \partial \) on weakly pseudoconvex manifolds. Trans. Am. Math. Soc.181, 273–292 (1973) · Zbl 0276.35071
[7] Lieb, I., Range, R.M.: Lösungsoperatoren für den Cauchy-Riemann-Komplex mitC k-Abschätzungen. Math. Ann.253, 145–164 (1980) · Zbl 0441.32007
[8] Øvrelid, N.: Generators of the maximal idelas of \(A(\bar D)\) . Pac. J. Math.39, 219–223 (1971) · Zbl 0231.46090
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.