Let $f\in L^p(\Omega),$ then does there exist a function $u$ on $\Omega \subset \mathbb{R}^n$ such that the weak derivative of $u$ is f?
In other words, does differential equation $u'=f$ admits a distributional solutions $u$ which is also a function on $\Omega$.
If such a solution $u$ exists, then does $u \in L^p(\Omega)$ always?
The fact that $f$ is in $L^p$ guarantees that $f$ is locally integrable. Indeed, if $K\subset\Omega$ is compact, then by Hölder $$ \int_K|f|=\int_\Omega |f|\,1_K\leq\|f\|_p\,|K|^{1/q}<\infty. $$
In dimension $1$, using that $f$ is locally integrable, we can define $$ u(x)=\int_{0}^{x}f. $$ Then for a test function $\phi$ we have $\def\abajo{\\[0.2cm]}$ \begin{align} \int_\Omega u\phi'&=\int_\Omega\phi'(x)\,\int_0^xf(t)\,dt\,dx\\ &=\int_{\mathbb R}f(t)\int_t^\infty\phi'(x)\,dx\,dt\\ &=-\int_{\mathbb R}f(t)\phi(t)\,dt=-\int_\Omega f\phi. \end{align}
Fubini is justified because $f$ is locally integrable and $\phi'$ has compact support, hence the integral occurs in a compact set where $f\,\phi'$ is integrable.
It's not obvious to me that this idea can be pushed to higher dimensions, where the natural generalization of the FTC is Stokes' theorem, and I wouldn't know (haven't thought much about this) how to deal with the region when using Fubini.