Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four

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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 L 2 -norm of curvature on a metric ball, we can find a smooth metric with bounded curvature which is C 1 , α -close to the original metric on a smaller ball but still of definite size.