We begin with the proof of (3.9).

The Bochner formula for $|\nabla u|^{1-\alpha}$ is given by

To estimate the left hand side, we observe that since ${\rm trace}(\nabla^{2}u)=\Delta u=0$, if follows if $\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $\nabla^{2}u$ then $\sum\lambda_{i}=0$. In particular, if $\lambda_{n}$ is the largest eigenvalue then by the Schwarz inequality,

This leads to the improved Kato inequality

where ${\rm v}$ is any vector with $|{\rm v}|=1$.

If rewrite

and apply the improved Kato inequality, then we get the Bochner formula

which gives nontrivial information for any $\alpha<\frac{1}{n-1}$. Namely,

As in the proof of (3.5), let $\varphi\geq 0$ be a smooth function as in [ChCo1], with ${\rm supp}\,\varphi\subset B_{2}(p)$, $|\nabla\phi|,|\Delta\phi|\leq C(n)$.

By multiplying both sides of (3.16) by $\varphi$ and integrating we obtain

Now we use that if $u$ is a harmonic $\delta$-splitting map then $|\nabla u|^{1-\alpha}$ is bounded and $\fint_{B_{2}(p)}\Big||\nabla u|^{1-\alpha}-\fint_{B_{2}(p)}|\nabla u|^{1-\alpha}\Big|$ is small. In particular, for $\delta$ sufficiently small, we have

which proves (3.9).

Now we proceed with the proof of (3.10).

We begin with some computations. Given $\omega^{\ell}=du^{1}\wedge\cdots\wedge du^{\ell}$ we have

In particular, if $u$ is a $\delta$-splitting map then

As in the proof of (3.5), let $\varphi\geq 0$ be a smooth function as in [ChCo1], with ${\rm supp}\,\varphi\subset B_{2}(p)$, $|\nabla\varphi|,|\Delta\phi|\leq C(n)$. Then we have

if $\delta(n,\epsilon)$ is sufficiently small. Thus, if $\delta=\delta(n,\epsilon)$ is sufficiently small,

This completes the proof of (3.10). ∎