We study the problem of extending functions from linear affine subvarieties for the Bergman scale of spaces on convex finite type domains. Our results solve the problem for H¹(D). For other Bergman spaces the result is ϵ-optimal.

The Hartogs Theorem for holomorphic functions is generalized in two settings: a CR version (Theorem 1.2) and a corresponding theorem based on it for $C^k$$\overline{\partial }$-closed forms at the critical degree, $0\le k\le \infty$ (Theorem 1.1). Part of Frenkel’s lemma in $C^k$ category is also...

We give a proof of an integral formula of Berndtsson which is related to the inversion of Fourier-Laplace transforms of $\overline{\partial }$-closed $(n,n-1)$-forms in the complement of a compact convex set in ${ℂ}^n$.

We study the residue current $R^f$ of Bochner-Martinelli type associated with a tuple $f=(f_1,\cdots ,f_m)$ of holomorphic germs at $0\in {\mathbf{C}}^n$, whose common zero set equals the origin. Our main results are a geometric description of $R^f$ in terms of the Rees valuations associated with the ideal $\left(f\right)$ generated by $f$ and a characterization of when the annihilator ideal of $R^f$ equals $\left(f\right)$.

We give a sufficient condition for a hermitian holomorphic vector bundle over the disk to be quasi-isometric to the trivial bundle. One consequence is a version of Cartan's lemma on the factorization of matrices with uniform bounds.

Our objective is to construct residue currents from Bochner-Martinelli type kernels; the computations hold in the non complete intersection case and provide a new and more direct approach of the residue of Coleff-Herrera in the complete intersection case; computations involve crucial relations with toroidal varieties and multivariate integrals of the Mellin-Barnes type.

A new representation of the Cauchy kernel ${}_{\Gamma }$ for an arbitrary acute convex cone Γ in ℝⁿ is found. The domain of holomorphy of ${}_{\Gamma }$ is described. An estimation of the growth of ${}_{\Gamma }$ near the singularities is given.

Dans cet article, nous nous proposons d’étudier le noyau, l’image et une éventuelle formule d’inversion de la transformation de Radon réelle dans les domaines linéairement concaves. Nous rappelons que, dans ${ℝ}^2$, on sait reconstruire une fonction à partir de sa transformation de Radon lorsque celle-ci est connue le long de toutes les droites de l’espace. Notre propos sera, en quelque sorte, d’établir une version semi-globale de ce résultat. Nous verrons ainsi que, modulo un noyau que nous préciserons,...

We study a division problem in the Hardy classes $H^p\left(\right)$ of the unit ball of ${ℂ}^2$ which generalizes the $H^p$ corona problem, the generators being allowed to have common zeros. MPrecisely, if S is a subset of , we study conditions on a ${ℂ}^k$-valued bounded Mholomorphic function B, with $B_{|S}=0$, in order that for 1 ≤ p < ∞ and any function $f\in H^p\left(\right)$ with $f_{|S}=0$ there is a ${ℂ}^k$-valued $H^p\left(\right)$ holomorphic function F with f = B·F, i.e. the module generated by the components of B in the Hardy class $H^p\left(\right)$ is the entire module $M_S:=f\in H^p\left(\right):f_{|S}=0$. As a special case,...