On the range of a class of complex Monge-Ampère operators on compact Hermitian manifolds

Let $(X,\omega )$ be a compact Hermitian manifold of complex dimension n. Let β be a smooth real closed $(1,1)$ form such that there exists a function $\rho \in \text{PSH}(X,\beta )\cap L^{\infty }\left(X\right)$. We study the range of the complex non-pluripolar Monge-Ampère operator $〈{(\beta +d{d}^c\cdot )}^n〉$ on weighted Monge-Ampère energy classes on X. In particular, when ρ is assumed to be continuous, we give a complete characterization of the range of the complex Monge-Ampère operator on the class $E(X,\beta )$, which is the class of all $\phi \in \text{PSH}(X,\beta )$ with full Monge-Ampère mass, i.e. ${\int }_X〈{(\beta +d{d}^c\phi )}^n〉={\int }_X{\beta }^n$.