# Estimates of $\Delta|\nabla u|$ for harmonic function $u$

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 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 $$\begin{split} &|\nabla u|\leqslant 1, \\ &\int_{B_{2}}||\nabla u|-1| < \epsilon\\ &\int_{B_{2}}|\nabla^{2}u|^{2p} < \epsilon \end{split}$$ for some integer $$p\geqslant1$$, then $$\int_{B_{1}}(\Delta|\nabla u|)^{p}<\delta\; ?$$ Note 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.

Yes, this inequality is true, and in fact the assumption about the smallness of $$D^2u$$ in $$L^{2p}$$ is not needed.

Using that $$|\nabla u|$$ is $$\epsilon$$-close to one in $$L^1(B_2)$$, we can find a point in $$B_1$$ where $$\nabla u$$ is $$C(n)\epsilon$$-close to the unit sphere, say $$-e_1$$ after a rotation. Then $$u_1 + 1$$ is nonnegative (since $$|\nabla u| \leq 1$$), harmonic, and $$C(n)\epsilon$$-close to zero somewhere in $$B_1$$. By the Harnack inequality, $$u_1$$ is pointwise $$C(n)\epsilon$$-close to $$-1$$ in $$B_{3/2}$$. Using again that $$|\nabla u| \leq 1$$ we conclude that $$\nabla u$$ is pointwise $$C(n)\sqrt{\epsilon}$$-close to $$-e_1$$ in $$B_{3/2}$$. The interior derivative estimates for harmonic functions then imply that $$|D^ku|$$ are pointwise $$C(n,k)\sqrt{\epsilon}$$-small in $$B_1$$ for $$k \geq 2$$. This completes the proof with $$\delta = C(n,p)\epsilon^{p}$$.

• Thank you for your answer. And my actual question is to obtain the estimate on a manifold with $|Ric|\leqslant\epsilon$. This method seems to not work and I have asked another question here mathoverflow.net/q/462340/520372 . Commented Jan 17 at 1:10