Potential theory in several quaternionic variables

Abstract

In this paper, we establish the quaternionic versions of the potential description of various "small" sets related to the quaternionic plurisubharmonic functions in $\mathbb{H}^n$. We use the quaternionic capacity introduced in \cite{wan4} to characterize the $(-\infty)$-sets of plurisubharmonic functions, as the sets of the vanishing capacity. The latter requirement is also equivalent to the negligibility of the set. We also prove the Josefson's theorem on the equivalence of the locally and globally quaternionic polar sets in $\mathbb{H}^n$, following the method of Bedford-Taylor.