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.
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
If $f\in L^1$, then by the Sobolev inequality it belongs to $W^{-1,n/(n-1)}$. By Calderon-Zygmund estimates, this implies that the solution belongs to $W^{1,n/(n-1)}$. You can then differentiate the equation and repeatedly apply this inequality to get your desired estimate (with improved integrability, now $W^{k,n/(n-1)}$ instead of $W^{k,1}$).
