Degenerate quarternionic Monge-Amp\`ere equations in weighted energy classes

In this paper, we consider degenerate quaternionic Monge-Amp\`ere 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-Amp\`ere 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-Amp\`ere equation, characterization of finite energy range of complex Monge-Amp\`ere 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-Amp\`ere operator in the end.