On the Range of a class of Complex Monge-Amp\`ere operators on compact Hermitian manifolds

Let (X,ω) 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 ρ ∈ PSH(X,β)∩ L∞(X). We study the range of the complex non-pluripolar Monge-Amp\`ere operator (β+ddᶜ·)ⁿ on weighted Monge-Amp\`ere 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-Amp\`ere operator on the class E(X,β), which is the class of all φ ∈ PSH(X,β) with full Monge-Amp\`ere mass, i.e. ∫X (β+ddᶜφ)ⁿ=∫Xβⁿ.