The Perron-Bremermann envelope for $q$-plurisubharmonic functions on unbounded domains in $\mathbb{C}^n$

Abstract

Let $D$ be an unbounded domain in $\mathbb{C}^n$, and let $f$ be a bounded continuous function prescribed on the boundary of $D$. We show that, if $D$ has $r$-peak points on its boundary and is of bounded type, $f$ extends to a maximal bounded continuous function $F$ on $\overline{D}$ that is $q$-plurisubharmonic and $(n-q-1)$-plurisuperharmonic (i.e., $(dd^c F)^n=0$) on $D$, and that coincides with the Perron-Bremermann envelope created with respect to bounded $q$-plurisubharmonic functions on $\overline{D}$.