Let $\Omega \subseteq \mathbb R^n$ be a bounded open set with smooth boundary. Let $k\geq 1$, $\alpha\in(0,1)$, $a_{ij},b_i,f \in C^{k,\alpha}(\Omega)$ for $i,j=1,...,n$, and define the operator

$$L = \sum_{i,j=1}^n a_{ij} \partial_{ij} + \sum_{i=1}^n b_i \partial_i.$$

Assume further that $L$ is uniformly elliptic, i.e. $\sum_{i,j} a_{ij}(x) \xi_i\xi_j\geq \lambda \|\xi\|^2$ for all $x \in \Omega$, $\xi\in\mathbb R^n$, and some $\lambda > 0$. Standard elliptic regularity theory ensures that the Dirichlet problem $L u =f$ in $\Omega$, with $u=0$ on $\partial\Omega$, admits a solution $u \in C^{k+2,\alpha}$, whose Holder norm depends only on that of the above data.

I was wondering whether there are any known conditions under which the regularity of the coefficients $b_i$ can be weakened from $C^{k,\alpha}$ to $C^{k-1,\alpha}$, while retaining the existence of a solution $u \in C^{k+2,\alpha}$. It is unclear to me whether the proof of Schauder's estimates based on reduction to constant-coefficient equations (e.g. Theorem 13.2.1. of Jost's text) can be adapted to this setting under any sensible conditions. (While my question is general, I will note that $f$ happens to be of class $ C^{k+2,\alpha}$ in my particular use case.)