I have this result from a book:
Assume $u$ is harmonic in $U$. Then $$|D^{\alpha} u(x_0)|\le \frac{C_k}{r^{n+k}}||u||_{L^1(B(x_0,r))}$$ for each ball $B(x_0, r)\subset U$ and each multiindex $\alpha$ of order $|\alpha|=k$, and $C_0 = \frac{1}{\alpha(n)}$, $C_k = \frac{(2^{n+1}nk)^k}{\alpha(n)}$
and this one that my teacher passed to us:
Let $\Omega\subset \mathbb{R}^n$ be open and let $f\in C^{\infty}(\Omega)$. Then these are equivalent:
1) $f$ is analytic in $\Omega$
2) Given any closed ball $B\subset\Omega$, there exists $C>0$ such that $$\sup_{x\in B}|\partial^{\alpha} f(x)|\le C^{|\alpha|+1}\alpha!$$
It looks like one is a special case of another. How do I get from this one to the other?
Also what is this norm $||u||_{L^1(B(x_0,r))}$?
