On relative $L^{\mathrm{\infty }}$ estimate for complex Monge-Ampère equations

We prove a relative $L^{\mathrm{\infty }}$ estimate for a class of complex Monge-Ampère type equations on Kähler manifolds. It provides a unified approach to Tundinger type estimate and uniform estimate. It also improves the previous results about modulus of continuity, stability estimates, and $W^{1,1}$-estimates of Green’s functions. The argument is based on the PDE method developed by Guo-Phong-Tong and constructing appropriate comparison metrics from entropy bound.