Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to $$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$.

If $ f \in L^1(\Omega)$ then one just misses $ u \in C(\Omega)$. There was a result of Wente that said something like if $ f= \nabla a \cdot \nabla^\perp b$ (where $a$ and $b$ have certain regularity assumptions, but not really enough to see the right hand side is better than $L^1$) then $ u \in C(\Omega)$. I believe there is also a result that says something like if $f$ in a certain Hardy space (I am not familiar with these spaces) then one also has $ u$ continuous.

QUESTION. I recall someone mentioning a version similar to the above. They had said if $ f(x) ={\rm div}(F(x))$ where $ F \in W^{1,1}(\Omega, R^2)$ then $ u \in C(\Omega)$. So my question is. Is this correct or not ?

Thanks