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ADDED: I can't resist adding an anecdote to this: Right after I learned the trick in the paragraph above from a paper of Schoen-Simon-Yau, I went to a colleague's office to show it to him. As it happens, Eli Stein was there, and he exclaimed, "But it's in my book!" And indeed it is. You will find it presented very nicely in VII.3.1 "A subharmonic property of the gradient" of Stein's 1970 book, "Singular Integrals and Differentiability Properties of Functions". It is obvious that S-S-Y did not know this or forgot, because their proof is much messier than Stein's.

$$ \|b\|_{q/2}, \|u\|_p < C $$

where $q > n$ for some $p > 1$ on $B(x,r)$, then there is a bound on $\|u\|_\infty$ on, say, $B(x,r/2)$.

$$ \|b\|_{q/2}, \|u\|_p < C, $$

where $q > n$, for some $p > 1$ on $B(x,r)$, then there is a bound on $\|u\|_\infty$ on, say, $B(x,r/2)$.

Combining the first two shows that if $u$ satisfies $-\Delta u \le cu^2$ and $ \|u\|_{n/2} $ is sufficiently small on $ B(x,r) $, then there is a bound on $ \|u\|_\infty $ on $ B(x,r/2) $.

In each application there is a curvature tensor $F$ that satisfies a PDE of the form $$ -\Delta F = Q(F), $$ where $Q$ depends quadratically on $F$. Moreover, there is a convergence theorem when there is a uniform pointwise bound on $ F $ (for Einstein manifolds you use the Cheeger-Gromov convergence theorem). Applying the results above to $u = |F|$ using coverings with smaller and smaller balls leads to a convergence theorem when there is a uniform bound on $ \|F\|_{n/2} $ where the convergence can fail at only a finite number of points (where in the limit there is too much of $ \|F\|_{n/2} $ for the estimates above to hold).

Combining the first two shows that if $u$ satisfies $-\Delta u \le cu^2$ and $ \|u\|_ {n/2} $ is sufficiently small on $ B(x,r) $, then there is a bound on $ \|u\|_\infty $ on $ B(x,r/2) $.

In each application there is a curvature tensor $F$ that satisfies a PDE of the form $$ -\Delta F = Q(F), $$ where $Q$ depends quadratically on $F$. Moreover, there is a convergence theorem when there is a uniform pointwise bound on $ F $ (for Einstein manifolds you use the Cheeger-Gromov convergence theorem). Applying the results above to $u = |F|$ using coverings with smaller and smaller balls leads to a convergence theorem when there is a uniform bound on $ \|F\|_ {n/2} $ where the convergence can fail at only a finite number of points (where in the limit there is too much of $ \|F\|_{n/2} $ for the estimates above to hold).