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 $\beta$ be a smooth real closed $(1,1)$ form such that there exists a function $\rho \in PSH(X,\beta )\cap L^{\mathrm{\infty }}(X)$. 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 $\rho$ is assumed to be continuous, we give a complete characterization of the range of the complex Monge-Ampère operator on the class $\mathcal{E}(X,\beta )$, which is the class of all $\phi \in PSH(X,\beta )$ with full Monge-Ampère mass, i.e. ${\int }_X⟨(\beta +d{d}^c\phi {)}^n⟩={\int }_X{\beta }^n$.