Do we have that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$. I am wondering if there is a way to see this that does not just involve invoking something such as this, i.e. a direct proof would be nice.
2026-04-22 22:26:27.1776896787
$L^p$-bounding inequality
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