The strong topology of ω-plurisubharmonic functions

Extremal Ω-plurisubharmonic Functions as Envelopes of Disc Functionals

For each closed, positive (1, 1)-current ω on a complex manifold X and each ω-upper semicontinuous function φ on X we associate a disc functional and prove that its envelope is equal to the supremum of all ω-plurisubharmonic functions dominated by φ. This is done by reducing to the case where ω has a global potential. Then the result follows from Poletsky’s theorem, which is the special case ω ...

Tangents of plurisubharmonic functions

Tangents of plurisubharmonic functions Local properties of plurisubharmonic functions are studied by means of the notion of tangent which describes the behavior of the function near a given point. We show that there are plurisubharmonic functions with several tangents, disproving a conjecture of Reese Harvey.

Local inequalities for plurisubharmonic functions

The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of polynomial, algebraic and analytic functions. The paper has much in common with an earlier paper [Br] of the author.

Weak and Strong Forms of ω-Continuous Functions

The Strong Oka’s Lemma, Bounded Plurisubharmonic Functions and the ∂̄-neumann Problem

The classical Oka’s Lemma states that if Ω is a pseudoconvex domain in C, n ≥ 2, then − log δ is plurisubharmonic where δ is some distance function to the boundary. Let M be a complex hermitian manifold with the metric form ω. Let Ω be relatively compact pseudoconvex domain in M . We say that a distance function δ to the boundary bΩ satisfies the strong Oka condition if it can be extended from ...