Title: The Calabi flow with rough initial data,
Abstract: In this paper, we prove that there exists a dimensional constant $\delta > 0$
such that given any background Kähler metric $\omega$, the Calabi flow with
initial data $u_0$ satisfying \begin{equation*} \partial \bar \partial u_0 \in
L^\infty (M) \text{ and } (1- \delta )\omega < \omega_{u_0} < (1+\delta
)\omega, \end{equation*} admits a unique short time solution and it becomes
smooth immediately, where $\omega_{u_0} : = \omega +\sqrt{-1}\partial
\bar\partial u_0$. The existence time depends on initial data $u_0$ and the
metric $\omega$. As a corollary, we get that Calabi flow has short time
existence for any initial data satisfying \begin{equation*} \partial \bar
\partial u_0 \in C^0(M) \text{ and } \omega_{u_0} > 0, \end{equation*} which
should be interpreted as a "continuous Kähler metric". A main technical
ingredient is Schauder-type estimates for biharmonic heat equation on
Riemannian manifolds with time weighted Hölder norms.