Can one show that the composition of two bounded linear operators $T_1, T_2: L_2(\Omega) \to H^1(\Omega)$ maps $L_2$ functions to $H^2$, i.e., $T_3 = T_1 T_2$ and $T_3: L_2(\Omega) \to H^2(\Omega)$?
What I tried to do is to bound the $H^2$ norm of a function $\varphi = T_1 T_2 u$. For that I tried to bound $\| \partial_{x_i} \partial_{x_j} \varphi\|_{L_2}$. I tried to do it for $u \in L_2$ and $u \in C^\infty$ (possibly coupled with an argument that $C^\infty$ is dense in $H^2$), but nothing seemed to lead to anywhere.
A second thing I tried to do, is estimating the operator norm of $\| D_{x_i} D_{x_j} T_1 T_2 \|_{L_2}$, where $D_{x_l}$ represents the differentiation operator. However, what I would need for that is that $D_{x_j}$ and $T_1$ commute within the operator norm, which I couldn't see why.