Le problème de Gleason est résolu dans le cas particulier des domaines analytiques réels pseudo-convexes de . Dans ce cas, les points faiblement pseudo-convexes peuvent former un sous-ensemble de dimension 2 du bord.
Le problème de Gleason est ramené à une question sur en montrant que l’ensemble des points de Kohn-Nirenberg a au plus une dimension. En fait, exception faite d’un sous-ensemble unidimensionnel, les points faiblement pseudo-convexes du bord sont des -points comme ceux étudiés par Range et admettent donc des estimations de par des normes de la borne supérieure locales.
The Gleason problem is solved on real analytic pseudoconvex domains in . In this case the weakly pseudoconvex points can be a two-dimensional subset of the boundary. To reduce the Gleason problem to a question it is shown that the set of Kohn-Nirenberg points is at most one-dimensional. In fact, except for a one-dimensional subset, the weakly pseudoconvex boundary points are -points as studied by Range and therefore allow local sup-norm estimates for .
@article{AIF_1983__33_2_77_0, author = {Fornaess, John Erik and Ovrelid, M.}, title = {Finitely generated ideals in $A(\omega )$}, journal = {Annales de l'Institut Fourier}, volume = {33}, year = {1983}, pages = {77-85}, doi = {10.5802/aif.916}, mrnumber = {84h:32019}, zbl = {0489.32013}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_1983__33_2_77_0} }
Fornaess, John Erik; Ovrelid, M. Finitely generated ideals in $A(\omega )$. Annales de l'Institut Fourier, Tome 33 (1983) pp. 77-85. doi : 10.5802/aif.916. http://gdmltest.u-ga.fr/item/AIF_1983__33_2_77_0/
[1] Hölder estimates for the Z-equation with a support condition, Pacific J. Math., 90 (1980), 249-257. | MR 82b:32029 | Zbl 0453.32006
,[2] Pseudoconvex domains : Existence of Stein neighbourhoods, Duke J. Math., 44 (1977), 641-662. | Zbl 0381.32014
and ,[3] The Mergelyan property for weakly pseudoconvex domains, Manuscripta Math., 22 (1977), 199-208. | MR 56 #15983 | Zbl 0391.32010
and ,[4] Finitely generated ideals in Banach algebras, J. Math. Mech., 13 (1964), 125-132. | MR 28 #2458 | Zbl 0117.34105
,[5] Approximation of functions in pseudoconvex domains and Leibenzon's theorem, Bull. Acad. Pol. Sci. Ser. Math. Astron. et Phys., 19 (1971), 37-42. | Zbl 0214.33701
,[6] Finitely generated ideals in certain function algebras, J. Funct. Anal., 7 (1971), 212-215. | MR 43 #929 | Zbl 0211.43902
and ,[7] Boundary behavior of Z on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542. | MR 48 #727 | Zbl 0256.35060
,[8] A pseudoconvex domain not admitting a holomorphic support function, Math. Ann., 201 (1973), 265-268. | MR 48 #8850 | Zbl 0248.32013
and ,[9] Die Cauchy-Riemannschen Differentialgleichung auf streng pseudokonveksen Gebieten : Stetige Randwerte, Math. Ann., 199 (1972), 241-256. | MR 48 #6468 | Zbl 0231.35055
,[10] Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 449-474. | Numdam | Zbl 0128.17101
,[11] Øn Hölder estimates for Zu = f on weakly pseudoconvex domains, Cortona Proceedings, Cortona, 1976-1977, 247-267. | Zbl 0421.32021
,[12] Generators of the maximal ideals of A (D), Pac. J. Math., 39 (1971), 219-233. | MR 46 #9393 | Zbl 0231.46090
,