I am looking for a reference or proof of a lemma (if it's true) or a counter-example otherwise. It goes as follows:


Let $B\_1$ and $B\_2$ are two concentric balls of radius $1$ and $2$ in some $n$-dimensional Euclidean space. Then for any $f$ and $F$, there exists a $C>0$ such that
$$
\text{if }\;\Delta f=\text{div}F\; \text{ weakly, then }\;
\|f\|\_{L^p\_1(B\_1)}\leq C\big(\|F\|\_{L^p(B\_2)}+\|f\|\_{L^p(B\_2)}\big).
$$



Reference or proof of a lemma in PDE
I would call this estimate "the classical Calderon-Zygmund estimate" but indeed it is hard to track down a statement for the right-side in divergence form. Usually it is stated as an estimate from $L^p$ to $W^{2,p}$, when what you want is $W^{-1,p}$ to $W^{1,p}$. This is unfortunate, because the latter is the more natural and useful/important estimate! These statements are of course related, and you can get from one to the other.


However, for a direct proof of the statement you want, the only place I know to find it is in [our book](https://link.springer.com/book/10.1007/978-3-030-15545-2): see Proposition 7.3 (which actually proves a slightly more general statement than the one you want). The arxiv version is [here](https://arxiv.org/pdf/1705.05300.pdf).


Note that the statement is of course only true for $1<p<\infty$, there are counterexamples given in our book as well.


