### $\overline{\partial}$-closed extension of CR-forms with singularities on a generic manifold.

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Let D ⊂ ℂⁿ and $G\subset {\u2102}^{m}$ be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂D (resp. ∂G) and let X be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Suppose in addition that the domain D (resp. G) is locally ² smooth on A (resp. B). We shall determine the “envelope of holomorphy” X̂ of X in the sense that any function continuous on X and separately holomorphic on (A×G)∪(D×B) extends to a function continuous on X̂ and holomorphic on the interior of X̂. A generalization of this result to N-fold...

Let $G={K}^{\u2102}$ be a complex reductive group. We give a description both of domains $\Omega \subset G$ and plurisubharmonic functions, which are invariant by the compact group, $K$, acting on $G$ by (right) translation. This is done in terms of curvature of the associated Riemannian symmetric space $M:=G/K$. Such an invariant domain $\Omega $ with a smooth boundary is Stein if and only if the corresponding domain ${\Omega}_{M}\subset M$ is geodesically convex and the sectional curvature of its boundary $S:=\partial {\Omega}_{M}$ fulfills the condition ${K}^{S}\left(E\right)\ge {K}^{M}\left(E\right)+k(E,n)$. The term $k(E,n)$ is explicitly computable...

We construct a domain of holomorphy in ${\u2102}^{N}$, N≥ 2, whose envelope of holomorphy is not diffeomorphic to a domain in ${\u2102}^{N}$.

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$$Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:\phantom{\rule{4pt}{0ex}}W:=(A\times Y)\cup (X\times B)\to Z$ extends to a holomorphic mapping $\widehat{f}$ on $\widehat{W}:=\left\{(z,w)\in X\times Y:\phantom{\rule{4pt}{0ex}}\tilde{\omega}(z,A,X)+\tilde{\omega}(w,B,Y)\<1\right\}$ such that $\widehat{f}=f$...

The aim of this paper is to present an extension theorem for (N,k)-crosses with pluripolar singularities.

Tuboids are tube-like domains which have a totally real edge and look asymptotically near the edge as a local tube over a convex cone. For such domains we state an analogue of Cartan’s theorem on the holomorphic convexity of totally real domains in ${\mathbb{R}}^{n}\subset {\u2102}^{n}$.

We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map φ such that the associated composition operator is not open onto its image.

Let X be a closed analytic subset of an open subset Omega of Rn. We look at the problem of extending functions from X to Omega.