Let F ∈ ℂ[x,y]. Some theorems on the dependence of branches at infinity of the pencil of polynomials f(x,y) - λ, λ ∈ ℂ, on the parameter λ are given.

We discuss an example of an open subset of a torus which admits a dense entire curve, but no dense Brody curve.

We show that a certain solution operator for ∂ in a space of forms square integrable against e-|z|2 is canonical, i.e., that it gives the minimal solution when applied to a ∂-closed form, and gives zero when applied to a form orthogonal to Ker ∂.As an application, we construct a canonical homotopy operator for i∂∂.

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family $y={\phi }_s(t,x)=sb₁\left(x\right)t+b₂\left(x\right)t²+\cdots$ of analytic curves in ℂ × ℂⁿ passing through the origin, $Con{v}_{\phi }\left(f\right)$ of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series $f({\phi }_s(t,x),t,x)$ converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that $E=Con{v}_{\phi }\left(f\right)$ if and only if...

One proves the density of an ideal of analytic functions into the closure of analytic functions in a $L^p\left(\mu \right)$-space, under some geometric conditions on the support of the measure $\mu$ and the zero variety of the ideal.

In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of Cn obtained by P. Hu and W. Zhang [2], and S. Li [4].

Let $V$ be an algebraic variety in $C^n$ and when $k\ge 0$ is an integer then $\mathrm{Pol}(V,k)$ denotes all holomorphic functions $f\left(z\right)$ on $V$ satisfying $\left|f\left(z\right)\right|\le C_f(1+|z|{)}^k$ for all $z\in V$ and some constant $C_f$. We estimate the least integer $ϵ(V,k)$ such that every $f\in \mathrm{Pol}(V,k)$ admits an extension from $V$ into $C^n$ by a polynomial $P(z_1,...,z_n)$, of degree $k+ϵ(V,k)$ at most. In particular ${lim}_{k\>\infty }ϵ(V,k)$ is related to cohomology groups with coefficients in coherent analytic sheaves on $V$. The existence of the finite integer $ϵ(V,k)$ is for example an easy consequence of Kodaira’s Vanishing Theorem.

The classical Mittag-Leffler theorem on meromorphic functions is extended to the case of functions and hyperfunctions belonging to the kernels of linear partial differential operators with constant coefficients.

This paper deals with atomic decomposition and factorization of functions in the holomorphic Hardy space $H^1$. Such representation theorems have been proved for strictly pseudoconvex domains. The atomic decomposition has also been proved for convex domains of finite type. Here the Hardy space was defined with respect to the ordinary Euclidean surface measure on the boundary. But for domains of finite type, it is natural to define $H^1$ with respect to a certain measure that degenerates near Levi-flat points...

In this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth...