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Under the usual condition that the volume of a geodesic ball is close to theEuclidean one or the injectivity radii is bounded from below, we prove a lowerbound of the $C^{\alpha} W^{1, q}$ harmonic radius for manifolds with boundedBakry-\'Emery Ricci curvature when the gradient of the potential is bounded.Under these conditions, the regularity that can be imposed on the metrics underharmonic coordinates is only $C^\alpha W^{1,q}$, where $q>2n$ and $n$ is thedimension of the manifolds. This is almost 1 order lower than that in theclassical $C^{1,\alpha} W^{2, p}$ harmonic coordinates under bounded Riccicurvature condition [And]. The loss of regularity induces some difference inthe method of proof, which can also be used to address the detail of $W^{2, p}$convergence in the classical case.  Based on this lower bound and the techniques in [ChNa2] and [WZ], we extendCheeger-Naber's Codimension 4 Theorem in [ChNa2] to the case where themanifolds have bounded Bakry-\'Emery Ricci curvature when the gradient of thepotential is bounded. This result covers Ricci solitons when the gradient ofthe potential is bounded.  During the proof, we will use a Green's function argument and adopt a linearalgebra argument in [Bam]. A new ingradient is to show that the diagonalentries of the matrices in the Transformation Theorem are bounded away from 0.Together these seem to simplify the proof of the Codimension 4 Theorem, even inthe case where Ricci curvature is bounded.

Bounds on harmonic radius and limits of manifolds with bounded Bakry-\'Emery Ricci curvature