Degenerate quarternionic Monge-Ampère equations in weighted energy classes

Abstract

In this paper, we consider degenerate quaternionic Monge-Ampère equations in weighted energy class ${\mathcal{E}}_{\chi }(\Omega )$ where $\Omega$ is a quarternionic domain in ${\mathbb{H}}^n$ and $\chi$ is a weight function which satisfies some natural conditions. Firstly we prove that the quaternionic Monge-Ampère operator is well-defined for functions in ${\mathcal{E}}_{\chi }(\Omega )$, in particular ${\mathcal{E}}_p(\Omega ),p>0$. Secondly, we prove that fine property holds in the Cegrell type class $\mathcal{E}(\Omega )$. As an application, we prove a mass concentration theorem for the quarternionic plurisubharmonic envelope. In the study of complex Monge-Ampère equation, characterization of finite energy range of complex Monge-Ampère operator was a central problem which aroused the interest of experts in the subject. As a quaternionic analogue, we prove a theorem which explicitly characterizes the finite energy range of \emph{quaternionic} Monge-Ampère operator in the end.