We prove the Hölder continuity for proper holomorphic mappings onto certain piecewise smooth pseudoconvex domains with "good" plurisubharmonic peak functions at each point of their boundaries. We directly obtain a quite precise estimate for the exponent from an attraction property for analytic disks. Moreover, this way does not require any consideration of infinitesimal metric.

Let $(X,\omega )$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X,\omega )$ of the complex Monge-Ampère operator acting on $\omega$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^p$-density belong to $(X,\omega )$ and proving that $(X,\omega )$ has the...

We point out relations between Siciak’s homogeneous extremal function ${\Psi }_B$ and the Cauchy-Poisson transform in case $B$ is a ball in ℝ². In particular, we find effective formulas for ${\Psi }_B$ for an important class of balls. These formulas imply that, in general, ${\Psi }_B$ is not a norm in ℂ².

Let $K$ be a compact subset of an hyperconvex open set $D\subset {ℂ}^n$, forming with D a Runge pair and such that the extremal p.s.h. function ω(·,K,D) is continuous. Let H(D) and H(K) be the spaces of holomorphic functions respectively on D and K equipped with their usual topologies. The main result of this paper contains as a particular case the following statement: if T is a continuous linear map of H(K) into H(K) whose restriction to H(D) is continuous into H(D), then the restriction of T to $H\left(D_{\alpha }\right)$ is a continuous...

The purpose of this paper is to present a concise survey of the main properties of biholomorphically invariant pluricomplex Green functions and to describe a number of new examples of such functions. A concept of pluricomplex geodesics is also discussed.

Let $X$ be a compact Kähler manifold and $\Delta$ be a $ℝ$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that $K_X+\Delta$ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \mathrm{Supp}\left(\Delta \right)$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta$. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.

Les fonctions plurisousharmoniques négatives dans un domaine Ω de ℂⁿ forment un cône convexe. Nous considérons les points extrémaux de ce cône, et donnons trois exemples. En particulier, nous traitons le cas de la fonction de Green pluricomplexe. Nous calculons celle du bidisque, lorsque les pôles se situent sur un axe. Nous montrons que cette fonction ne coïncide pas avec la fonction de Lempert correspondante. Cela donne un contre-exemple à une conjecture de Dan Coman.

We prove, among other results, that $min(u,v)$ is plurisubharmonic (psh) when $u,v$ belong to a family of functions in $\mathrm{psh}\left(D\right)\cap {\Lambda }_{\alpha }\left(D\right),$ where ${\Lambda }_{\alpha }\left(D\right)$ is the $\alpha$-Lipchitz functional space with $1<\alpha <2.$ Then we establish a new characterization of holomorphic functions defined on open sets of ${ℂ}^n.$

We generalize a theorem of Siciak on the polynomial approximation of the Lelong class to the setting of toric manifolds with an ample line bundle. We also characterize Lelong classes by means of a growth condition on toric manifolds with an ample line bundle and construct an example of a nonample line bundle for which Siciak's theorem does not hold.

The main result of this paper is the following: if a compact subset E of ${ℝ}^n$ is UPC in the direction of a vector $v\in S^{n-1}$ then E has the Markov property in the direction of v. We present a method which permits us to generalize as well as to improve an earlier result of Pawłucki and Pleśniak [PP1].