# $L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature

This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ . The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $$(M^{n},g)$$, if $$u$$ is a harmonic function in $$B_{2}(x)\subset (M^{n},g)$$, then $$\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\nabla^{2}u|^{2}-|\nabla|\nabla u||^{2}=|\nabla u|\Delta|\nabla u|-Ric(\nabla u,\nabla u)\leqslant |\nabla^{2}u|^{2}.$$ 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 $$(M^{n},g)$$ is a manifold with $$|Ric|\leqslant\epsilon$$ and $$u$$ is a harmonic function in $$B_{2}(x)\subset (M^{n},g)$$ 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 $$\int_{B_{1}}(\Delta|\nabla u|)^{p}<\delta \ ?$$ This problem has been solved in the case of Euclidean space. See Estimates of $\Delta|\nabla u|$ for harmonic function $u$ . But on a manifold with $$|Ric|\leqslant\epsilon$$, that method seems to not work, since we cannot get harmonicity for $$\nabla u$$ and apply Harnack inequality.

Also, it is easy when $$p=1$$, since we can multiply the Cheeger-Colding’s cutoff function (which has bounded laplacian provided lower bounds on Ricci, see https://doi.org/10.2307/2118589) and then integrate by parts. But for integer $$p\geqslant2$$, I can’t see how to do it.

• The function $|\nabla u|$ is subharmonic, this should be the key. Commented Jan 17 at 3:05