My question is about the regularity of a metric whose curvature is bounded in $L^2$. Of course, this question doesn't really make sense since the regularity of the metric depends on the coordinates choice.
But, thanks to Deturck-Kazdan, we know that harmonic coordinates are the best choice, more precisely, if $g\in C^2$ and $Ric\in C^{k,\alpha}$ for $k>0$ in harmonic coordinates then $g\in C^{k+2,\alpha}$.
The proof use some "classical elliptic theory" and the fact that in harmonic coordinates we get roughly speaking $\Delta g=Ric$.
My question is two folds.
1) Assuming the existence of harmonic coordinates, do we have an $L^p$-estimate, like $Ric \in L^p$ then $g\in W^{2,p}$ for $p>1$. It seems reasonable with respect to regularity theory but I didn't find any reference.
2) let assume 1) with $p=2$ and $n=4$(dimension of the manifold). Then $g\in W^{2,2}$ but this space doesn't not belongs to $C^0$, hence it seems difficult to give sense to any harmonic coordinates since the coefficients of the Laplace-Beltrami won't be regular enough to apply the regularity theory. Can we give some sense to this assuming only $Rm$ or $Ric$ in $L^2$.
My motivation comes from the regularity theory for critical point of quadratic functional with respect to the curvature in dimension $4$, like $$\int_M \vert Ric_g\vert^2\, dv_g$$, for instance Einstein metric are critical point of this functional and they are known to be analytic (theorem 5.26 of Besse) assuming that the manifold is $C^1$ and I guess the metric at least $C^0$ before bootstraping. Can we give a sense to all this theory working only in energy space?
