Let $\Omega \subset R^n $ ($n \geq 2$) a domain with smooth boundary. Consider an arbitrary ball $B(x_0,R) \Subset \Omega$. I know that for a function in $L^{p}(\Omega)$ with $1<p<2$ it happens
$$ \int_{B(x_0,r)} |u - u_{x_0,r}|^p \leq C\int_{B(x_0,R)} |u - u_{x_0,R}|^p$$ for all $0<r \leq R$ where $C$ is a positive constant that depends on $n$ and $p$ and $u_{x_0,r} = \frac{\int_{B(x_0,r)} u}{|B(x_0,R)|}$ Probably this happens if $p \geq 2$. Please, someone could point me a reference? I am searching for it, but I am not finding...
Thanks for your attention