Some properties of the functions of the form $v\left(x\right)={\sum }_{i=0}^m{\left|x\right|}^i{h}_i\left(x\right)$ in ℝⁿ, n ≥ 2, where each $h_i$ is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

This article provided some sufficient or necessary conditions for a class of integral operators to be bounded on mixed norm spaces in the unit ball.

We consider a class of maximal plurisubharmonic functions and prove several properties of it. We also give a condition of maximality for unbounded plurisubharmonic functions in terms of the Monge-Ampère operator $\left(d{d}^c{e}^u\right)ⁿ$.

Let $X$ be a compact complex nonsingular surface without curves, and $E$ a holomorphic vector bundle of rank 2 on $X$. It turns out that the associated projective bundle $\mathbf{P}E$ has no divisors if and only if $E$ is “strongly” irreducible. Using the results concerning irreducible bundles of [Banica-Le Potier, J. Crelle, 378 (1987), 1-31] and [Elencwajg- Forster, Annales Inst. Fourier, 32-4 (1982), 25-51] we give a proof of existence for bundles which are strongly irreducible.

Let $(𝒮,0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_i$, $i\in I$. The Nash map associates to each irreducible component $C_k$ of the space of arcs through $0$ on $𝒮$ the unique component of $E$ cut by the strict transform of the generic arc in $C_k$. Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E·E_i\<0$ for any $i\in I$.

Let $X$ be a complex space of dimension $n$, not necessarily reduced, whose cohomology groups $H^1(X,𝒪),...,H^{n-1}(X,𝒪)$ are of finite dimension (as complex vector spaces). We show that $X$ is Stein (resp., $1$-convex) if, and only if, $X$ is holomorphically spreadable (resp., $X$ is holomorphically spreadable at infinity). This, on the one hand, generalizes a known characterization of Stein spaces due to Siu, Laufer, and Simha and, on the other hand, it provides a new criterion for $1$-convexity.

In this paper we study generic coverings of ${ℂ}^2$ branched over a curve s.t. the total space is a normal analytic surface, in terms of a graph representing the monodromy of the covering, called monodromy graph. A complete description of the monodromy graphs and of the local fundamental groups is found in case the branch curve is $\{x^n=y^m\}$ (with $n\le m$) and the degree of the cover is equal to $n$ or $n-1$.

For each 2-dimensional complex torus $T$, we construct a compact complex manifold $X\left(T\right)$ with a ${ℂ}^2$-action, which compactifies ${\left({ℂ}^{*}\right)}^4$ such that the quotient of ${\left({ℂ}^{*}\right)}^4$ by the ${ℂ}^2$-action is biholomorphic to $T$. For a general $T$, we show that $X\left(T\right)$ has no non-constant meromorphic functions.

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain K. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta-Siciak extremal function). When the inscribed ellipse method...

We compute the constant sup $(1/degP)\left(ma{x}_{S}log\right|P|-{\int }_{S}log|P\left|d\sigma \right)$ : P a polynomial in ${ℂ}^n$, where S denotes the euclidean unit sphere in ${ℂ}^n$ and σ its unitary surface measure.