Let $\Omega\subset\mathbb R^2$ be an open set, and let $f\in L^1_{loc}(\Omega)$ have a weak derivative $f_{xx}\in L^1_{loc}(\Omega)$.
- Does this imply the existence of $f_x\in L^1_{loc}(\Omega)$?
- If not, does the existence of all second-order weak derivatives (or just $f_{xx}$ and $f_{xy}$) imply it?
- Or maybe suffices the existence of $f_{xx}$, but assuming more regularity (for example $L^p_{loc}(\Omega)$, or even $C^{0,\alpha}$)?
Things that I know:
- Existence of highly regular $f_{xy}$ is not sufficient
- If $f_x$ and $f_y$ exist continuous then $f\in C^1(\Omega)$, but I don't know whether it is related
- Does the existence of weak derivatives require the lower order derivatives also to exist?
