Global Regularity for the∂-b-Equation onCRManifolds of Arbitrary Codimension

LetMbe aC∞compactCRmanifold ofCR-codimensionl≥1andCR-dimensionn-lin a complex manifoldXof complex dimensionn≥3. In this paper, assuming thatMsatisfies conditionY(s)for someswith1≤s≤n-l-1, we prove anL2-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for(0,s)-forms onM.

SUR L'ÉQUATION DE CAUCHY–RIEMANN TANGENTIELLE DANS UNE CALOTTE STRICTEMENT PSEUDOCONVEXE

We search a cohomological and a geometrical characterization of the open subsets of a strictly pseudoconvex boundary in a Stein manifold on which one can solve the tangential Cauchy–Riemann equation in all bidegrees. On cherche une caractérisation cohomologique et géométrique des ouverts du bord d'un domaine strictement pseudoconvexe relativement compact d'une variété de Stein sur lesquels on peut résoudre l'équation de Cauchy–Riemann tangentielle en tout bidegré.

On holomorphic extension of functions on singular real hypersurfaces inℂn

The holomorphic extension of functions defined on a class of real hypersurfaces inℂnwith singularities is investigated. Whenn=2, we prove the following: everyC1function onΣthat satisfies the tangential Cauchy-Riemann equation on boundary of{(z,w)∈ℂ2:|z|k<P(w)},P∈C1,P≥0andP≢0, extends holomorphically inside provided the zero setP(w)=0has a limit point orP(w)vanishes to infinite order. Furthermore, ifPis real analytic then the condition is also necessary.