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, 2π), we show that locally defined weak solutions (C^1,1-solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coordinates. This shows the weak Kähler-Einstein metrics constructed by Guenancia-Paun, and independently by Yao, are all actually strong-conical Kähler-Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical Kähler-Ricci flat metrics defined over Cⁿ, which depends on a Calderon-Zygmund theory in the conical setting. The regularity of globally defined weak-conical Kähler-Einstein metrics is already proved by Guenancia-Paun using a different method.