There exists $C=C(n,{\rm v},q)$ such that if $M^{n}$ satisfies $|{\rm Ric}_{M^{n}}|\leq n-1$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$. then for each $q<2$,

$\displaystyle\fint_{B_{1}(p)}|{\rm Rm}|^{q}\leq C\,.$

(1.6)

If in addition, $M^{n}$ is assumed to be Einstein, then for every $q<2$ we have that

$\displaystyle{\rm Vol}(T_{r}(\{x\in B_{1}(p):r_{x}\leq r\}))\leq C\,r^{2q}$

(1.7)