Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four

We obtain a local smoothing result for Riemannian manifolds with bounded Ricci curvatures in dimension four. More precisely, given a Riemannian metric with bounded Ricci curvature and small L²-norm of curvature on a metric ball, we can find a smooth metric with bounded curvature which is C1,α-close to the original metric on a smaller ball but still of definite size.