Interior elliptic regularity in W^{k,1} spaces In the context of a second-order linear elliptic PDE with smooth coefficients for a function $u: \mathbb{R}^n \to \mathbb{R}$, the interior $W^{k,p}$ regularity theorems I have seen in the literature apply only if $1<p<\infty$. I am interested in the case $p=1$.
Here is a simple example (essentially Theorem B.3.2 in McDuff/Salamon: $J$-holomorphic curves and symplectic topology), the PDE $\Delta u = \text{div}X$ for an $L^p$ vector field $X=(f_1,\dots,f_n)$ on $\Omega\subseteq\mathbb{R}^n$:

Theorem. Let $1<p<\infty$, let $\Omega\subseteq\mathbb{R}^n$ be open. Let $u\in L^1_{\text{loc}}(\Omega,\mathbb{R})$
  and $f_1,\dots,f_n\in L^p_{\text{loc}}(\Omega,\mathbb{R})$
  satisfy $$ \int_{\Omega}u(x)\Delta\phi(x)\textrm{d}x =
-\sum_{i=1}^n\int_{\Omega}f_i(x)\partial_i\phi(x)\textrm{d}x $$ for all $\phi\in C^\infty_0(\Omega,\mathbb{R})$. Then $u\in W^{1,p}_{\text{loc}}(\Omega,\mathbb{R})$.

Is this still true for $p=1$? If not, what would be a counterexample?
[I have edited the rest of the question.]
If you prefer, pick your favourite integers $k\geq0$ and $n\geq2$ and prove or disprove that every $u\in W^{k,1}_{\text{loc}}(\mathbb{R}^n,\mathbb{R})$ with $\Delta u\in W^{k,1}_{\text{loc}}(\mathbb{R}^n,\mathbb{R})$ lies in $W^{k+2,1}_{\text{loc}}(\mathbb{R}^n,\mathbb{R})$.
(As proved in Ornstein 1962, the elliptic estimate $||u||_{W^{2,p}} \leq C(||u||_{L^p} +||\Delta u||_{L^p})$ fails for $p=1$. This suggests that there exists a $u\in L^1_{\text{loc}}$ with $\Delta u\in L^1_{\text{loc}}$ and $u\notin W^{2,1}_{\text{loc}}$, but it is not obvious to me how to show that. Another thing that fails for $p=1$ is the surjectivity of $\Delta: W^{2,p}\to L^p$; see 2.1 in Bourgain/Brezis 2002 for an even stronger statement.)
If (as I expect) $u\in W^{k,1}_{\text{loc}}$ and $\Delta u\in W^{k,1}_{\text{loc}}$ do not imply $u\in W^{k+2,1}_{\text{loc}}$, what is the best regularity of $u$ we can deduce in general?
For instance, the Sobolev embedding $W^{k,1}_{\text{loc}} \subset W^{k-1,n/(n-1)}_{\text{loc}}$ and the $p>1$ regularity theory imply $u\in W^{k+1,n/(n-1)}_{\text{loc}}$. I expect that the same idea with fractional Sobolev spaces works as well and yields $u\in W^{k+2-\varepsilon,n/(n-\varepsilon)}_{\text{loc}}$ for every $\varepsilon>0$. Unfortunately I have not found references for the theorems needed for this conclusion. Is it true? Can one get even slightly more regularity?
Where are all these questions discussed in the literature?
 A: The problem is indeed coming form the fact that singular integrals, such as the Hilbert transform, although bounded on $L^p$ for $1<p<+\infty$ are failing to be bounded on $L^1$ or $L^\infty$. 
However, a good substitute for $L^1(\mathbb R^n)$ could be the Hardy space $\mathcal H^1(\mathbb R^n)$, defined as the subspace of $u\in L^1(\mathbb R^n)$ such that
$$
R_j u\in L^1(\mathbb R^n),\quad\text{where $ R_j $ is the Fourier multiplier $\xi_j/\vert \xi\vert$.}
$$
Then the singular integral $\Delta^{-1}\nabla \text{div}$ should send 
$\mathcal H^1(\mathbb R^n)$ into itself. Note that it is a non-trivial task to localize this to a proper open subset $\Omega$ of $\mathbb R^n$, since the Hardy space is not a local space: $u\in \mathcal H^1(\mathbb R^n)$ does NOT imply $\chi u\in \mathcal H^1(\mathbb R^n)$ for a smooth compactly supported $\chi$ (in fact, a function $u\in \mathcal H^1(\mathbb R^n)$ must have a zero  integral, typically a non-local condition).
Last but not least, the 1D Hilbert transform $\mathfrak h$
is the Fourier multiplier $\text{sign } \xi$ and with $u(x)=e^{-π x^2}$, you find
$$
(\text{Fourier}(\mathfrak h u))(\xi)=(\text{sign } \xi) e^{-π \xi^2}
$$ 
which is not in Fourier$(L^1(\mathbb R))$ since it is discontinuous (at 0), implying that $\mathfrak h u$ does not belong to $L^1(\mathbb R).$ 
