Me and a friend of mine are working on the following exercise: let $\Omega$ be a bounded domain of $\mathbb{R}^n$, characterize the sets: $$ X=\{ D_{x_i} u : u \in L^2(\Omega) ; \ i=1, \dots n \} \qquad Y=\{ D_{x_i x_j} v : v \in H^1(\Omega); \ i,j=1, \dots n \}$$
We started with the case $n=1$ and $\Omega=(0,1) $ for simplicity and concluded that $X=Y=H^{-1}(0,1)$ essentially through the following steps (here $D(A)$ is a short notation to indicate the set of weak derivatives of functions of $A$):
(i) $D(\mathcal{D}(0,1))=\{ \phi \in \mathcal{D}(0,1) : \int_{[0,1]} \phi =0 \}$
(ii) $ D(H^1_0(0,1))=\{ f \in L^2(0,1) : \int_{[0,1]} f = 0 \}$
(iii) $ D(H^1(0,1))= L^2(0,1) $
(iv) $D(L^2(0,1))= H^{-1}(0,1)$
where we mostly used elementary techniques (density results, classical inequalities...). Later on we discovered that (iii) may be proved very easily using FCT and the characterization $H^1(0,1)=\{ u \in AC([0,1]) : u' \in L^2(0,1) \}$.
We are asking ourselves if this result is actually true and if it can be generalized to higher dimensions without being forced to complicated and technical proofs.