Interior elliptic regularity in W^{k,1} spaces

Question

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?

Answer 1

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 $L^1(\mathbb R).$