Let $L$ be an elliptic operator of the form $$ Lu := (-1)^m \sum_{|\alpha|=2m} a_\alpha(x) D^\alpha u + \sum_{|\alpha|\leq 2m-1} b_\alpha (x) D^\alpha u $$ with smooth coefficients and $u$ defined in a smooth bounded domain of $R^N$, $N \geq 2m$. Are there some kind of Schauder type estimates for solutions of the problem $$ L u = f(x), $$ coupled with some boundary conditions (for example Dirichlet or Navier boundary conditions).

As far as I know such kind or results exist but for operator of the form $$ L u = (-\Delta)^m u + \sum_{|\alpha| \leq 2m - 1} c_\alpha(x) D^\alpha u. $$