Where can I find a proof of the following fact?



> 
> If
> $$w(u,x\_{0},r)=\sup \_{B\_{r}(x\_{0})}u-\inf \_{B\_{r}(x\_{0})}u$$
> for some function $u(x)$ satisfies
> $$ w\left(u,x\_{0},{\tfrac {r}{2}}\right)\leq \lambda w\left(u,x\_{0},r\right)$$
> for a fixed $0 < \lambda < 1$ and all sufficiently small values of $r$, then $u$ is Hölder continuous.
> 
> 
> 



Oscillation and Hölder continuity
Just prove it yourself:


Take $r=1$. Then
$$w(x\_0,2^{-n})\leq \lambda^n w(x\_0,1)=:C\lambda^n.$$
To estimate $|u(x\_0)-u(y)|$, where $y$ is close to $x\_0$, choose $n$ so that
$|x\_0-y|\in[2^{-n-1},2^{-n}].$
Then
$$|u(x\_0)-u(y)|\leq w(x\_0,2^{-n})\leq C\lambda^{n}=C. 2^{-hn}\leq 2^hC\_1|x\_0-y|^h,$$
where $h=-\log\lambda/\log2$.


