I am reading this classic to learn elliptic equations.
Corollary 9.10 on pp. 235 states for $u\in W_0^{2,p}(\Omega)$, $1<p<\infty$: First, $$ (9.33) \quad \|D^2 u\|_p\le C(n,p) \|\Delta u\|_p $$ and, if $p=2$, $$ (9.34) \quad \|D^2 u\|_2= \|\Delta u\|_2. $$
The book says the corollary follows immediately from Theorem 9.9 (Calderon-Zygmund inequality).
I do not quite understand how (9.33)-(9.34) follow from Theorem 9.9, which had established $\|D^2w\|_p\le C\|f\|_p$ for $w$ as a Newtonian potential, i.e., $w=N f$. We may try to match $\Delta u$ in (9.33) to $f$ in Theorem 9.9, but now $u$ in Corollary 9.10 seems not being the Newtonian potential itself. In other words, the form of $u$ in Theorem 9.9 does not cover the case of Corollary 9.10.
I can use integration by parts (as used in the proof of Theorem 9.9 for the case of $p=2$) to get (9.34), but have no idea how to prove (9.33). Did I miss some point in the book? Another reference to dig?