Calderon-Zygmund inequality on $W^{k,p}$

Question

Let $\Omega$ be a bounded region of $\mathbb R^n$, we have the Calderon-Zygmund inequality, i.e. $\|\nabla^2u\|_{L^p}\leq C\|\Delta u\|_{L^p}$.

Q For $k\geq1$, can we have $$\|\nabla^2u\|_{W^{k,p}}\leq C\|\Delta u\|_{W^{k,p}}$$

Answer 1

Yes. Since weak partial derivatives commute, we have $$ \|\nabla^2 u\|_{W^{k,p}}^p=\sum_{|\alpha|\leq k}\|\partial^\alpha\nabla^2u\|_{L^p}^p=\sum_{|\alpha|\leq k}\|\nabla^2 \partial^\alpha u\|_{L^p}^p\leq C^p\sum_{|\alpha|\leq k}\|\Delta \partial^\alpha u\|_{L^p}^p=C^p\|\Delta u\|_{W^{k,p}}^p. $$