Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, define
$\Omega\_{\rho}:=\{x\in\Omega:d(x,\partial\Omega)<\rho\}$, and let $f\in
L^{p}(\Omega)$ for some $p>N$. My question is:

Is it true that there exists $\rho\_{0}>0$ small such that $\left\vert f(x)\right\vert \leq Cd(x,\partial\Omega)^{\alpha
}$ $a.e. x \in$ $\Omega\_{\rho\_{0}}$, for some $C>0$ and some
$\alpha>-\frac{1}{p}$ ? If not, is it valid for some other $\alpha$ ?

Thanks in advance!
Uriel