Can the weak derivative be more regular than the function itself? Let $\Omega \subset \mathbb R^n$ be a bounded, open set.
Let $u \in L^1_{\text{loc}}(\Omega)$, and let $D_i u$ be its $i$-th weak partial derivative.
Is it possible that $D_i u \in L^2(\Omega)$ for every $i$, but $u \notin L^2(\Omega)$?

Apparently you can characterize the open sets for which this fails, as those who fail to have a version of Poincare's inequality. See Definition 5.2 and Theorem 5.3 on page 328 of this paper of Lions-Deny here.
Warning: Notation in that paper is old. So if you're diving in:

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*$\mathcal{E}_{L^2}^1(\Omega)$ is what we now call $W^{1,2}(\Omega)$. Similarly $\mathcal{E}_{q,p}^1(\Omega)$ is the space of functions in $L^q(\Omega)$ with gradients in $L^p(\Omega)$.

*$BL(\Omega)$ is the space of distributions in $\Omega$ whose gradients are in $L^2(\Omega)$. More generally $BL(E)$ is the space of distributions whose gradients lie in $E\subset \mathcal{D}'(\Omega)$.

*$BL^*(\Omega)$ is $BL(\Omega)$ modulo constants.

