Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following regularity estimates hold for $p=1$?


$$||u||\_{W^{k+1,p}(\Omega)}\leq C(||f||\_{W^{k,p}(\Omega)}+||u||\_{L^p(\Omega)}),$$


for some constant independent of $u$? This is true for $p>1$, but I am not finding any reference for $p=1$. Any hint would be appreciated. 



$W^{k,1}$ regularity for elliptic equations
In the case of the Laplace operator and $k=0$, see Lemma 14 in <https://arxiv.org/pdf/0809.2172.pdf>. The argument should work for more general divergence elliptic equation in the divergence form and higher order derivatives. See also <https://mathoverflow.net/a/298962/121665>


