The complex Monge-Ampère type equation on compact Hermitian manifolds and applications

We prove the existence and uniqueness of continuous solutions to the complex Monge-Ampère type equation with the right hand side in Lp, p>1, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi [17,18] to compact Hermitian manifolds which a priori are not in the Fujiki class. These generalisations lead to a number of applications: we obtain partial results on a conjecture of Tosatti and Weinkove [40] and on a weak form of a conjecture of Demailly and Paun [11].