If I recall correctly, if we only know apriori that, say, $u\in L^1(\Omega)$ for some nice bounded domain $\Omega$ and we denote $u_h$ a sequence (as $h\to 0$) of regularizations of $u$, and we can show that $\partial_i u_h$ converges in $L^1$ to some function $v\in L^1$, then the conclusion is that $\partial_i u$ (with the derivative originally taken in the distributional sense) is in fact in $L^1$ as well (and hence $u\in W^{1,1}$). I think this is well established.
Now what if we're in a situation where we can show that all the second derivatives of $u_h$ (actual derivatives, since $u_h$ is smooth) converge to functions in $L^1$. Does it follow from this that in fact $u\in W^{2,1}$? The point is that nothing is said apriori about the first derivatives of $u_h$.
Edit: I just realized that a simple application of Gagliardo Nirenberg interpolation between the 0th and 2nd derivatives of $u_h$ would give us that the first derivatives $\partial_i u_h$ are cauchy in $L^1$ and give us an affirmative answer for my question. Is this correct?