There exists $C=C({\rm v})$ such that if $M^{4}$ satisfies $|{\rm Ric}_{M^{4}}|\leq 3$ and ${\rm Vol}(B_{1}(p))>{\rm v}>0$, then

$\displaystyle\fint_{B_{1}(p)}|{\rm Rm}|^{2}\leq C\,.$

(1.8)

Furthermore, we have the sharp weak-$L^{2}$ estimate on the harmonic radius,

$\displaystyle{\rm Vol}(T_{r}(\{x\in B_{1}(p):r_{h}\leq r\}))\leq Cr^{4}\,.$

(1.9)

If we assume in addition that $M^{4}$ is Einstein, then the same result holds with the harmonic radius $r_{h}$ replaced by the regularity scale $r_{x}$.