We describe compact subsets K of ∂𝔻 and ℝ admitting holomorphic functions f with the domains of existence equal to ℂ∖K and such that the pluripolar hulls of their graphs are infinitely sheeted. The paper is motivated by a recent paper of Poletsky and Wiegerinck.

Let $M_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in $M_{m,n}$ is >br>    $R_{m,n}=Z\in M_{m,n}:I^{\left(m\right)}-ZZ*ispositivedefinite$. Here $I^{\left(m\right)}\in M_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L_{\alpha }^p\left(R_{m,n}\right)$ is defined to be the space $L^p{R}_{m,n};{\left[det(I^{\left(m\right)}-ZZ*)\right]}^{\alpha }d{\mu }_{m,n}\left(Z\right)$, where ${\mu }_{m,n}$ is the Lebesgue measure in $M_{m,n}$, and $H_{\alpha }^p\left(R_{m,n}\right)\subset L_{\alpha }^p\left(R_{m,n}\right)$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if $Re\beta >(\alpha +1)/p-1$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then     $f\left(\right)=T_{m,n}^{\beta }\left(f\right)\left(\right),\in R_{m,n},$where $T_{m,n}^{\beta }$ is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p <...

Every homogeneous circular convex domain $D\subset {\mathbb{C}}^n$ (a bounded symmetric domain) gives rise to two interesting Lie groups: The semi-simple group $G=Aut\left(D\right)$ of all biholomorphic automorphisms of $D$ and its isotropy subgroup $K\subset GL\left(n,\mathbb{C}\right)$ at the origin (a maximal compact subgroup of $G$). The group $G$ acts in a natural way on the compact dual $X$ of $D$ (a certain compactification of ${\mathbb{C}}^n$ that generalizes the Riemann sphere in case $D$ is the unit disk in $\mathbb{C}$). Various authors have studied the orbit structure of the $G$-space $X$, here we are interested...

We prove a sufficient condition for products of Toeplitz operators $T_f{T}_{ḡ}$, where f,g are square integrable holomorphic functions in the unit ball in ℂⁿ, to be bounded on the weighted Bergman space. This condition slightly improves the result obtained by K. Stroethoff and D. Zheng. The analogous condition for boundedness of products of Hankel operators $H_{f}H{*}_g$ is also given.

We study hypersurfaces of complex projective manifolds which are invariant by a foliation, or more generally which are solutions to a Pfaff equation. We bound their degree using classical results on logarithmic forms.

In rings ${\Gamma }_M$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M={\left(M_l\right)}_{l\ge 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in ${\Gamma }_M$ such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to ${\Gamma }_M$, provided ${\Gamma }_M$ is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...

Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p\&gt;2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1\&lt;p\&lt;3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of ...

These are the notes from a one-week course on Braid Monodromy of Algebraic Curves given at the Université de Pau et des Pays de l’Adour during the Première Ecole Franco-Espagnole: Groupes de tresses et topologie en petite dimension in October 2009.This is intended to be an introductory survey through which we hope we can briefly outline the power of the concept monodromy as a common area for group theory, algebraic geometry, and topology of projective curves.The main classical results are stated...