Complex Monge-Amp\`ere equation in Orlicz space and Diameter Bound

In this paper, we establish diameter bounds for compact K\"ahler manifolds equipped with K\"ahler metrics $\omega$, assuming the associated measure lies in a specific Orlicz space and satisfies an integrability condition. Firstly, we prove a priori estimates for solutions of the complex Monge-Amp\`ere equation in Orlicz spaces, encompassing $L^{\infty}$ and stability estimates. This is achieved by employing Ko{\l}odziej's approach \cite{Ko98} and the argument of Guo-Phong-Tong-Wang \cite{GuPhToWa21}, respectively. Secondly, building on the work of Guo-Phong-Song-Sturm \cite{GuPhSoSt24-1}, we derive the uniform (local/global) estimates of the Green's function and its gradient for the associated K\"ahler metric $\omega$.