Interior derivative estimate for a harmonic function
= Interior derivative estimate for a harmonic function
If $u$ is harmonic on $B_r(x_0)$, then for every integer $m\geq1$,
$$
|D^mu(x_0)|\leq C(n,m)r^{-m}\sup_{B_r(x_0)}|u|.
$$
The first-order estimate follows by differentiating the ball mean-value formula and applying the divergence theorem; iteration gives higher orders.