Let us consider the equation in the 2d torus $$\nabla \cdot f=g^2,$$ where $\nabla \cdot f$ denotes the divergence operator and $g^2$ has zero mean. In particular, we can see these functions as they were defined in $\mathbb{R} ^2$ and they were $2 \pi-$ periodic in each variable.
I know that this equation doesn't admit a solution $f \in W^{1,1}$ if $g \in L^2$. My question is: Is there a solution in $W^{1,1}$ for every $g^2$ in the Lorentz space $L^{2,1}$? If this is not possible, What is the best regularity that we can expect for the solution given $g$ under this condition?