Suppose the partials of $u, v :\Bbb R^2\to \Bbb R$ exist and satisfy the Cauchy-Riemann equations everywhere, with $u$ and $v$ locally integrable. Is it true that $u$ and $v$ are also weak solutions? I know about Looman-Menchoff, but this theorem requires continuity (or boundedness) of $u+iv$, so I am seeking a generalization where we only assume $u+iv$ locally in $L^1$. Note that if $u$ and $v$ were weak solutions, this would follow from elliptic regularity.
It seems that all we need is to show that we need to integrate by parts, but the weakest requirement for this is that the partials are in $L^1$ (by Theorem 7.21 in Rudin's RCA). In other words, we would require $u$ and $v$ to be absolutely continuous, but I am not sure if this follows from the CR equations. Is the generalization of Looman-Menchoff true or am I missing a counter-example?