On the regularity problem of complex Monge-Ampere equations with conical singularities

In the category of metrics with conical singularities along a smooth divisor with angle in $(0,2\pi )$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the K"ahler-Einstein equations actually possess maximum regularity, which means the metrics are actually H"older continuous in the singular polar coordinates. This shows the weak K"ahler-Einstein metrics constructed by Guenancia-Paun \cite{GP}, and independently by Yao \cite{GT}, are all actually strong-conical K"ahler-Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical K"ahler-Ricci flat metrics defined over ${\text{C}}^n$, which depends on a Calderon-Zygmund theory in the conical setting.