Let $H^{-1}$ be dual space of $H_0^1$. Define the subspace $A = \{ u \in H^{-1}: Du \in H^{-1}\}$ and give it a norm $\|u\|_A^2 = \|u\|_{H^{-1}}^2 + \|Du\|_{H^{-1}}^2$ (or similar). I want to show this space $A$ is equal to $L^2$ and the norms are equivalent. On $\mathbb{R}^n, \mathbb{T}^n$ it is immediate by fourier transform. But I would like to show it on a bounded domain $\Omega$, for simplicity a bounded interval. (Note $Du$ denotes the distributional derivative).
One direction is trivial since $D: L^2 \rightarrow H^{-1}$ and $I: L^2 \rightarrow H^{-1}$ are bounded we have $L^2 \subset A$ with $\|u\|_A \leq 2\|u\|_{L^2}$.
On the other hand, since $Du \in H^{-1}$ then we know by Riesz representation that there is an $f\in H_0^1$ such that $Du = f - D^2f$ with $\|Du\|_{H^{-1}}^2 = \|f\|_{H^1}^2 = \|f\|_{L^2}^2 + \|Df\|_{L^2}^2$. My idea is to go integrate $Du$, thus $u = \int_0^x f(t)dt - Df + c$ for some constant $c$ (which depends on $u$). Since $f\in H_0^1$ and the interval is bounded, this is in $L^2$. Thus $A \subset L^2$. Now $$\|u\|_{L^2}^2 \leq C(\|\int_0^x f(t)dt\|_{L^2}^2 + \|Df\|_{L^2}^2 + \|c\|_{L^2}^2) \leq C(\|f\|_{L^2}^2 + \|Df\|_{L^2}^2 + c^2) \leq C(\|Du\|_{H^{-1}}^2+c^2)$$ Here I am stuck. How can I bound the constant $c$ by either $\|u\|_{H^{-1}}$ or $\|Du\|_{H^{-1}}$ ? Perhaps there is a better approach. I hope to eventually extend the result to higher dimensions and more derivatives.
What you want is called Necas inequality. See this post An inequality of J. Necas There are a couple of references to books where you can find the proof. In particular Necas, J. Direct Methods in the Theory of Elliptic Equations, Springer, 2012, p.186-190.