Rafał Czyż

An Inequality for the Beta Function with Application to Pluripotential Theory

We prove in this paper an inequality for the beta function, and we give an application in pluripotential theory.

The Geometry of m-Hyperconvex Domains

We study the geometry of m-regular domains within the Caffarelli–Nirenberg–Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every m-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.

On the Błocki–Zwonek conjectures

Let be a bounded pseudoconvex domain in , and let be the pluricomplex Green function with pole at in . It was conjectured by Błocki and Zwonek that the function given by is convex. Here is the Lebesgue measure in . In this note we give an affirmative answer to this conjecture when is biholomorphic to the unit ball or to the polydisc in , .

A Counterexample to a Conjecture by Błocki–Zwonek

ABSTRACT For a bounded pseudoconvex domain and pluricomplex Green function gΩ(z, a) with pole at a ∈ Ω, it was conjectured by Błocki and Zwonek that β(t) = logu2009λn({z ∈ Ω: gΩ(z, a) < t}) is a convex function on ( − ∞, 0). With Ω the annulus the Green function gΩ(z, a) with pole at a = 1 + 0i can be explicitly given in terms of Jacobi theta functions. We show numerically that in this case β is not convex.

Extension and approximation of m-subharmonic functions

Abstract Let be a bounded domain, and let f be a real-valued function defined on the whole topological boundary . The aim of this paper is to find a characterization of the functions f which can be extended to the inside to a m-subharmonic function under suitable assumptions on . We shall do so using a function algebraic approach with focus on m-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of m-subharmonic functions.

Kolodziej's subsolution theorem for unbounded pseudoconvex domains

In this paper we generalize Kolodziejs subsolution theorem to bounded and unbounded pseudoconvex domains, and in that way we are able to solve complex Monge-Ampere equations on general pseudoconve ...