The Gleason problem is solved on real analytic pseudoconvex domains in ${\mathbf{C}}^2$. In this case the weakly pseudoconvex points can be a two-dimensional subset of the boundary. To reduce the Gleason problem to a $\bar{\partial }$ 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 $\bar{\partial }$.

Si illustrano alcuni sviluppi della teoria delle foliazioni di Monge-Ampère e delle sue applicazioni alla classificazione delle varietà complesse non compatte.

We give a homotopy formula for the $\partial {̅}_b$ operator on a domain with (q+k)-concave boundary. As a consequence we show that the dimension of the $\partial {̅}_b$-cohomology groups for some CR manifolds q-concave at infinity is finite.

Dans cet article, on construit tout d’abord un noyau de Cauchy explicite dans la boule unité $B$ de ${\mathbf{C}}^n$ dont les valeurs au bord sont égales au noyau de Szegö. Puis, à partir de ce noyau, on construit explicitement les noyaux qui fournissent les solutions de l’équation $\bar{\partial }u=f$ qui sont orthogonales aux fonctions holomorphes dans les espaces $L^2(d{\sigma }_{\alpha })$, où $d{\sigma }_{\alpha }\left(z\right)=(1-|z{|}^2\left)d\lambda \right(z)$, $d\lambda \left(z\right)$ étant la mesure de Lebesgue et $\alpha$ un réel $\>-1$. Nous donnons ensuite les principales estimations dedans et au bord que vérifient ces solutions. Dans une deuxième...

We prove that any positive function on ℂℙ¹ which is constant outside a countable $G_{\delta }$-set is the order function of a fundamental solution of the complex Monge-Ampère equation on the unit ball in ℂ² with a singularity at the origin.