Finitely generated ideals in A(ω)
Annales de l'Institut Fourier, Tome 33 (1983) no. 2, pp. 77-85.

Le problème de Gleason est résolu dans le cas particulier des domaines analytiques réels pseudo-convexes de C 2 . 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 R-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 C 2 . 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 R-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},
     pages = {77--85},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {33},
     number = {2},
     year = {1983},
     doi = {10.5802/aif.916},
     mrnumber = {84h:32019},
     zbl = {0489.32013},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.916/}
}
TY  - JOUR
AU  - Fornaess, John Erik
AU  - Ovrelid, M.
TI  - Finitely generated ideals in $A(\omega )$
JO  - Annales de l'Institut Fourier
PY  - 1983
SP  - 77
EP  - 85
VL  - 33
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.916/
DO  - 10.5802/aif.916
LA  - en
ID  - AIF_1983__33_2_77_0
ER  - 
%0 Journal Article
%A Fornaess, John Erik
%A Ovrelid, M.
%T Finitely generated ideals in $A(\omega )$
%J Annales de l'Institut Fourier
%D 1983
%P 77-85
%V 33
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.916/
%R 10.5802/aif.916
%G en
%F 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) no. 2, pp. 77-85. doi : 10.5802/aif.916. http://www.numdam.org/articles/10.5802/aif.916/

[1] F. Beatrous, Hölder estimates for the Z-equation with a support condition, Pacific J. Math., 90 (1980), 249-257. | MR | Zbl

[2] K. Diederich and J. E. Fornæss, Pseudoconvex domains : Existence of Stein neighbourhoods, Duke J. Math., 44 (1977), 641-662. | Zbl

[3] J. E. Fornæss and A. Nagel, The Mergelyan property for weakly pseudoconvex domains, Manuscripta Math., 22 (1977), 199-208. | MR | Zbl

[4] A. Gleason, Finitely generated ideals in Banach algebras, J. Math. Mech., 13 (1964), 125-132. | MR | Zbl

[5] G. M. Henkin, Approximation of functions in pseudoconvex domains and Leibenzon's theorem, Bull. Acad. Pol. Sci. Ser. Math. Astron. et Phys., 19 (1971), 37-42. | Zbl

[6] N. Kerzman and A. Nagel, Finitely generated ideals in certain function algebras, J. Funct. Anal., 7 (1971), 212-215. | MR | Zbl

[7] J. J. Kohn, Boundary behavior of Z on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom., 6 (1972), 523-542. | MR | Zbl

[8] J. J. Kohn and L. Nirenberg, A pseudoconvex domain not admitting a holomorphic support function, Math. Ann., 201 (1973), 265-268. | MR | Zbl

[9] I. Lieb, Die Cauchy-Riemannschen Differentialgleichung auf streng pseudokonveksen Gebieten : Stetige Randwerte, Math. Ann., 199 (1972), 241-256. | MR | Zbl

[10] S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 449-474. | Numdam | Zbl

[11] M. Range, Øn Hölder estimates for Zu = f on weakly pseudoconvex domains, Cortona Proceedings, Cortona, 1976-1977, 247-267. | Zbl

[12] N. Øvrelid, Generators of the maximal ideals of A (D), Pac. J. Math., 39 (1971), 219-233. | MR | Zbl

Cité par Sources :