Weak partial derivatives lower the Sobolev order

Regularity bookkeeping. Since $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$ and $\mathrm{\mid }\beta \mathrm{\mid },\mathrm{\mid }\alpha +\beta \mathrm{\mid }\le k$, the classes $D^{\alpha }u$, $D^{\beta }u$ and $D^{\alpha +\beta }u$ are defined and lie in $L^p(\mathrm{\Omega };\mathbb{K})$ by [F1]; representatives of $L^p$ classes are locally integrable on the open set $\mathrm{\Omega }$, so both are classes in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega };\mathbb{K})$ to which the weak-differentiation calculus of [F2] applies.

Commutation. With $\alpha ,\beta$ as in the Given, the hypotheses of [F3] are met by step 1.1: $D^{\alpha }u$, $D^{\beta }u$ and $D^{\alpha +\beta }u$ exist in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$, hence $D^{\alpha }(D^{\beta }u)$ exists and satisfies $D^{\alpha }(D^{\beta }u)=D^{\alpha +\beta }u$ almost everywhere on $\mathrm{\Omega }$.

Descent of the order. By step 2.1, for every multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k-\mathrm{\mid }\beta \mathrm{\mid }$ the weak derivative $D^{\alpha }(D^{\beta }u)$ exists and equals $D^{\alpha +\beta }u$, which lies in $L^p(\mathrm{\Omega };\mathbb{K})$ by step 1.1; also $D^{\beta }u\in L^p(\mathrm{\Omega };\mathbb{K})$. By [F1] this says exactly that the class $D^{\beta }u$ belongs to $W^{k-\mathrm{\mid }\beta \mathrm{\mid },p}(\mathrm{\Omega };\mathbb{K})$. The identity $D^{\alpha }(D^{\beta }u)=D^{\alpha +\beta }u$ is an almost-everywhere identity of classes, and by uniqueness [F4] it is independent of the representatives chosen for $D^{\beta }u$ and $D^{\alpha +\beta }u$.