Characterizations of boundary pluripolar hulls

Abstract

We present some basic properties of the so called boundary relative extremal function and shed some light on Sadullaev's question about behavior of different kinds of extremal functions. We introduce and discuss boundary pluripolar sets and boundary pluripolar hulls. For B-regular domains the boundary pluripolar hull is always trivial on the boundary of the domain. We present a "boundary version" of Zeriahi's theorem on the completeness of pluripolar sets.2010 Mathematics Subject Classification. 32U05.

Local geodesics for plurisubharmonic functions

“…This differs from the definition of small unbounded locus used in [60], where the requirement was pluripolarity of L u , that is, existence of a function V ≡ −∞ which is psh in a neighborhood of D and V (ζ) = −∞ for all ζ ∈ L u . The present definition does not change L u ∩ D, while it allows the boundary part L u ∩ ∂D to be much bigger than pluripolar; such sets are called b-pluripolar [37]. For example, a compact set K ⊂ ∂D ⊂ C is b-pluripolar if and only if it is of zero Lebesgue measure.…”

Boundary relative extremal functions

“…Our goal in this section is to generalize Theorem 2.11 in [2] and solve Sadullaev's problem for circular sets in circular, strongly star shaped, (hence balanced) domains.…”

“…The answer apparently depends on the geometry and convexity properties of D and the choice of the compact set A ⊂ ∂D. For instance we showed in [2] that Sadullaev's question has a positive answer when D is a smooth pseudoconvex Reinhardt domain and A is multi-circular. The result in [2] exploits the relation between relative extremal functions and convex functions in a Reinhardt domain.…”

“…We denote the open unit disc in C by D, its boundary by T and the unit ball in C n (n ≥ 2) by B. Some basic properties of the boundary relative extremal function are given in [2,4,5,8,10] ( [4] appeared as [3] but the preprint is more relevant). Depending on the way the boundary is approached, plurisubharmonic function may have different boundary values.…”

An example concerning Sadullaev’s boundary relative extremal functions

“…Sadullaev [5] introduced several so-called boundary relative extremal functions for compact sets K in the boundary of domains D ⊂ C n , and asked whether their regularizations are perhaps always equal. Recently Djire and the author [1,2] gave a positive answer in certain cases where D and K are particularly nice.…”