  The regularity theory for pluriclosed flow hinges on obtaining $C^{\alpha}$
regularity for the metric assuming uniform equivalence to a background metric.
This estimate was established in \cite{StreetsPCFBI} by an adaptation of ideas
from Evans-Krylov, the key input being a sharp differential inequality
satisfied by the associated `generalized metric' defined on $T \oplus T^*$. In
this work we give a sharpened form of this estimate with a simplified proof. To
begin we show that the generalized metric itself evolves by a natural curvature
quantity, which leads quickly to an estimate on the associated Chern
connections analogous to, and generalizing, Calabi-Yau's $C^3$ estimate for the
complex Monge Ampere equation.
