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 $\C^{n}$, which depends on a Calderon-Zygmund theory in the conical setting.