Assume $M$ is a manifold. Assume $M$ is covered by domains $B_i$ and $\phi_i: B_i\to B_1(0)\subset{\mathbb R}^n$ are harmonic coordinates.

The Laplacian operator under a harmonic coordinate has a very simple form, $$ \Delta u = g^{ij} u_{ij}. $$ Assume the metric $g$ has a $C^{k, \alpha}$ bound under such coordinates. Then since components of the transition functions are harmonic by definition, by Schauder theory there should be a $C^{k+2, \alpha}$ bound on the transition functions $\phi_i\phi_j^{-1}$ - one takes advantage of the fact that derivatives of $g$ do not appear in the Laplacian.

My question is, why in many texts (Petersen's book, articles by Anderson and others), for the definition of $C^{k, \alpha}$ harmonic radius, it only requires (or deduce?) the transition functions to be $C^{k+1, \alpha}$. Did I miss something?