# $W^{k,1}$ regularity for elliptic equations

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

• Is there any good reason that why my question is not answered in a textbook? It sounds like a standard classical result. Apr 15, 2019 at 3:36

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 $$W^{k,1}$$).