I actually have have $\Omega \subseteq \mathbb{R}^{d_1+d_2}$ and set multiindex set $$\mathcal{A}:= \{\alpha = (\alpha_1, \alpha_2)\in \mathbb{N}^{d_1+d_2} : 2|\alpha_1| + |\alpha_2|\leq2 \}.$$ And space $$\mathrm{W}^{(1,2), p}(\Omega ; \mathbb{C}):=\left\{u \in \mathrm{L}^p(\Omega ; \mathbb{C}):(\forall {\alpha} \in \mathcal{A}) \partial^{{\alpha}} u \in \mathrm{L}^p(\Omega ; \mathbb{C})\right\}.$$ With norm $$\|u\|_{\mathrm{W}^{(1,2), p}(\Omega)}:=\sqrt[p]{\sum_{{\alpha} \in \mathcal{A}}\left\|\partial^\alpha u\right\|_{\mathrm{L}^p(\Omega)}^p}.$$
I'd like to show that differential operator $D: \mathrm{W}^{1, p} \rightarrow \mathrm{W}^{0, p}$ has a closed image so I can apply that to differential operator from $\mathrm{W}^{(1,2), p}$ to $\mathrm{W}^{(0,2), p}$.
Note that in general I do not know (yet) if $\mathrm{W}^{(1,2), p}$ is Banach so I am trying to evade solutions that use that fact (with operator being bounded from below).
