I am under the impression that the information is scattered all over the standard references, so I really hope to make sure that I understand correctly.
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with $C^1$ boundary.
Then, for each $1 < p < \infty$, does there exist some $C_p>0$ depending only on $p$ such that \begin{equation} \lVert D^2 u \rVert_p \leq C_p \Bigl( \lVert \Delta u \rVert_p +\lVert u \rVert_p \Bigr) \end{equation} for all $u \in W^{2,p}(\Omega)$? Here $D^2 u := \sum_{i,j=1}^n \partial_i\partial_j u$
I am pretty sure this is correct, but cannot find an exact theorem to derive this estimate from. Her
Moreover, how about the case $p=1$? Most references exclude or put restrictions on the $L^1$ case, so I would like to check as well.
Could anyone please help me?
This is a generalization of Schauder estimate which states that $$\|D^2 u\|_p \le C(p, n) \|\Delta u\|_p$$ for $1 < p < \infty$ and $u \in W^{2}_0(\Omega)$. The reason why we only consider $1 < p < \infty$ relies on some Calderon-Zygmund operator associated with the Fourier transform of $D^2u$ which is bounded from $L^p \to L^p$ only for these values of $p$. You can find more details about the proof here.
Note that the case $p = 2$ is a straightforward application of integration by parts. The generalisation for $p \in (1, \infty)$ requires more work.