We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in ${ℝ}^N$ is Hölder continuous then E admits a Szegö type inequality with weight function $dist{(x,\partial E)}^{-(1-\kappa )}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

We study Cegrell classes on compact Kähler manifolds. Our results generalize some theorems of Guedj and Zeriahi (from the setting of surfaces to arbitrary manifolds) and answer some open questions posed by them.

For algebraic surfaces, several global Phragmén-Lindelöf conditions are characterized in terms of conditions on their limit varieties. This shows that the hyperbolicity conditions that appeared in earlier geometric characterizations are redundant. The result is applied to the problem of existence of a continuous linear right inverse for constant coefficient partial differential operators in three variables in Beurling classes of ultradifferentiable functions.

We define directional Robin constants associated to a compact subset of an algebraic curve. We show that these constants satisfy an upper envelope formula given by polynomials. We use this formula to relate the directional Robin constants of the set to its directional Chebyshev constants. These constants can be used to characterize algebraic curves on which the Siciak-Zaharjuta extremal function is harmonic.

On a finite intersection of strictly pseudoconvex domains we define two kinds of natural Nevanlinna classes in order to take the growth of the functions near the sides or the edges into account. We give a sufficient Blaschke type condition on an analytic set for being the zero set of a function in a given Nevanlinna class. On the other hand we show that the usual Blaschke condition is not necessary here.

It is shown that there exist $C^{\infty }$ functions on the boundary of the unit disk whose graphs are complete pluripolar. Moreover, for any natural number k, such functions are dense in the space of $C^k$ functions on the boundary of the unit disk. We show that this result implies that the complete pluripolar closed $C^{\infty }$ curves are dense in the space of closed $C^k$ curves in ℂⁿ. We also show that on each closed subset of the complex plane there is a continuous function whose graph is complete pluripolar.

Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that ${\Gamma }_g\left(F\right):=(z,g(z\left)\right):z\in F$ is complete pluripolar in ${ℂ}^{N+1}$. Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that ${\Gamma }_g\left(D\right)$ is complete pluripolar in ${ℂ}^{N+1}$. These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86]...

The energy class ${}_p$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of ${ℱ}_p$ and its pluricomplex p-energy is proved.

Let D be a domain in ℂⁿ. The plurisubharmonic envelope of a function φ ∈ C(D̅) is the supremum of all plurisubharmonic functions which are not greater than φ on D. A bounded domain D is called c-regular if the envelope of every function φ ∈ C(D̅) is continuous on D and extends continuously to D̅. The purpose of this paper is to give a complete characterization of c-regular domains in terms of Jensen measures.

Let V be an analytic variety in a domain Ω ⊂ ℂⁿ and let K ⊂ ⊂ V be a closed subset. By studying Jensen measures for certain classes of plurisubharmonic functions on V, we prove that the relative extremal function ${\omega }_K$ is continuous on V if Ω is hyperconvex and K is regular.

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

We discuss the existence of the current $g^{k}T$, $k\in \mathbb{N}$ for positive and closed currents $T$ and unbounded plurisubharmonic functions $g$. Furthermore, a new type of weighted Lelong number is introduced under the name of weight $k$ Lelong number.