Capacities associated to the Siciak extremal function
Cegrell classes on compact Kähler manifolds
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.
Characterization of global Phragmén-Lindelöf conditions for algebraic varieties by limit varieties only
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.
Classes de Nevanlinna sur une intersection d'ouverts strictement pseudoconvexes.
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.
Comparison of Two Capacities in Cn.
Complete locally pluripolar sets.
Complete pluripolar curves and graphs
It is shown that there exist 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 functions on the boundary of the unit disk. We show that this result implies that the complete pluripolar closed curves are dense in the space of closed 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.
Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions [Book]
Composition of Potentials with Inner Functions.
Construction de disques analytiques et régularité de fonctions holomorphes au bord.
Continuation of positive, closed currents and analytic sets.
Convergence in capacity
We prove that if in Cₙ-capacity then . This result is used to consider the convergence in capacity on bounded hyperconvex domains and compact Kähler manifolds.
Convexité de l'espace des cycles
Convexity of sublevel sets of plurisubharmonic extremal functions
Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.
Croissance des fonctions plurisousharmoniques en dimension infinie
Les ensembles polaires dans , c’est-à-dire les ensembles où une fonction plurisousharmonique qui n’est pas identiquement admet cette valeur, apparaissent comme des ensembles exceptionnels dans beaucoup de problèmes en analyse complexe. Par exemple, la croissance d’une fonction plurisousharmonique en une variable quand une autre variable est fixée est essentiellement la même pour tout sauf quand appartient à un ensemble polaire. Dans l’article un résultat très précis et général de cette...