Suppose \(\Omega\subseteq \mathbb{R}^d\) is open and let \(u\in C^2(\Omega)\) be a harmonic function . Then for all balls \(B_r(x)\subseteq \Omega\) and every multi-index \(\alpha\subseteq \mathbb{N}^d\) the following estimate hold
\[ \lvert D^\alpha u(x)\rvert\le \frac{C(d,k)}{r^{d+k}}\int_{B_r(x)} \lvert u\rvert. \]
Proof
Proof by induction. For \(\lvert \alpha\rvert=0\) this is a consequence of the mean value property
.
Let \(\lvert \alpha\rvert=1\). Then \(\delta^\alpha u\) is harmonic and we again may apply the mean value property (on a smaller ball eventually). Using the divergence theorem
we estimate the expression with a supremum of \(u\) which we may estimate with the previous case.
The rest follows by induction.