I have a question concerning the so called excess of a function, which is often seen in regularity theory. It is defined as
$E(f,x,r):= \frac{1}{| B_r(x)|}\int_{B_r(x)} | f(y)-[f]_{x,r}|^2 dy$,
where $[f]_{x,r}$ denotes the mean value over $B_r(x)$. Let's say $f\in L^{\infty}(\Omega)$, $\Omega$ opend and bounded, and $B_r(x)\subset\subset \Omega$.
My question is now: if $f(x)\neq \infty$, is there a uniform constant $c>0$ (may depend on $f$, $x$) and a radius $r_0>0$ such that
$E(f,x,r) \leq c$ for all $r\leq r_0$?
Or even better for my purpose: Is there a constant $c>0$ and radius $r_0>0$ such that for every $0<r\leq r'\leq r_0$ it holds
$E(f,x,r)\leq c E(f,x,r')$?
I couldn't get a proof for one of those inequalities since the mapping $E$ is quite difficult to handle. Thanks in advance for your help.