  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.

Differential Geometry;Analysis of PDEs