Suppose that a function $u : \mathbb R^n \rightarrow \mathbb R$ is locally integrable.
Suppose furthermore that it has a weak derivative $w : \mathbb R^n \rightarrow \mathbb R^n$, that is, the $\nabla u := w$ has locally integrable components and we have
$$ -\int_{\mathbb R^n} w \phi dx = \int_{\mathbb R^n} u\partial_i \phi dx $$
for all compactly supported smooth functions $\phi : \mathbb R^n \rightarrow \mathbb R$.
If $u$ is in the Lebesgue space $L^p$ for some $1 \leq p \leq \infty$, what do we know about the possible Lebesgue space of its weak derivative $w$?
Does $u \in L^p$ and weakly differentiable automatically imply that $\nabla u \in L^q$ for some $q \in [1,\infty]$? Or can it happen (for some $p$) that $\nabla u$ is merely locally integrable?