On the Range of a class of Complex Monge-Amp\`ere 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 $\rho \in \mbox{PSH}(X,\beta)\cap L^{\infty}(X)$. We study the range of the complex non-pluripolar Monge-Amp\`ere operator $\langle(\beta+dd^c\cdot)^n\rangle$ on weighted Monge-Amp\`ere 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-Amp\`ere operator on the class $\mathcal E(X,\beta)$, which is the class of all $\varphi \in \mbox{PSH}(X,\beta)$ with full Monge-Amp\`ere mass, i.e. $\int_X\langle (\beta+dd^c\varphi)^n\rangle=\int_X\beta^n$.