For every $q<4$ there exists $C=C(n,{\rm v},q)$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq n-1$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$ and $u:B_{2}(p)\to\mathds{R}$ satisfies

$$|u|\leq 1\,,$$

$$|\Delta u|\leq 1\,,$$

then for every $q<4$

$\displaystyle\fint_{B_{1}(p)}|\nabla^{2}u|^{q}\leq C\,.$

(7.16)