arXiv subjects

S. Benelkourchi

Publications and source records attributed to S. Benelkourchi.

3 recordsLinked to original sources

arXiv2008

A priori estimates for weak solutions of complex Monge-Ampère equations

Let $X$ be a compact Kähler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal E}_χ(X,\om)$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends U.Cegrell's and S.Kolodziej's results and puts them into a unifying frame. It also gives a simple proof of S.T.Yau's celebrated a priori ${\mathcal C}^0$-estimate.

math.CV↗

arXiv2008

Plurisubharmonic functions with weak singularities

We study the complex Monge-Ampère operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Ampère energy. They generalize the classes introduced by U.Cegrell, and give a stratification of the space of (almost) all unbounded plurisubharmonic functions. We give an interpretation of these classes in terms of the speed of decreasing of the Monge-Ampère capacity of sublevel sets and solve associated complex Monge-Ampère equations.

math.CV↗

arXiv2004

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball $\B \sub \C^n$ with its relative logarithmic capacity in $\C^n$ with respect to the same ball $\B$. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of $\C^n$ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on $\C^n$ as well as the Cegrell class of \psh functions of bounded Monge-Ampère mass on a hyperconvex domain $\W \Sub \C^n.$ Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.

math.CV↗