The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\nabla^{2}u|^{2}-|\nabla|\nabla u||^{2}= |\nabla u|\Delta|\nabla u|\leqslant |\nabla^{2}u|^{2}$.
It $$ \frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\nabla^{2}u|^{2}-|\nabla|\nabla u||^{2}= |\nabla u|\Delta|\nabla u|\leqslant |\nabla^{2}u|^{2}. $$ It seems that there is a certain connection between $\Delta|\nabla u|$ and $|\nabla^{2}u|^{2}$. My question is how to obtain an analogous $L^{p}$ estimate for $\Delta|\nabla u|$. Let me fully state my question below.
For any $\delta>0$, does there exist $\epsilon>0$ such that if $u$ is a harmonic function in $B_{2}\subset\mathbb{R}^{n}$ with $|\nabla u|\leqslant1$, $\int_{B_{2}}||\nabla u|-1|<\epsilon$, and $\int_{B_{2}}|\nabla^{2}u|^{2p}<\epsilon$ $$ \begin{split} &|\nabla u|\leqslant 1, \\ &\int_{B_{2}}||\nabla u|-1| < \epsilon\\ &\int_{B_{2}}|\nabla^{2}u|^{2p} < \epsilon \end{split} $$ for some integer $p\geqslant1$, then $\int_{B_{1}}(\Delta|\nabla u|)^{p}<\delta$ ?
Note $$ \int_{B_{1}}(\Delta|\nabla u|)^{p}<\delta\; ? $$ Note that we only need the estimate in a smaller ball, so one may use a cutoff function. And it is easy when $p=1$, since we can multiply a good cutoff function with bounded laplacian and then integrate by parts. But for integer $p\geqslant2$, I can’t see how to do it.
