Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$ is continuous) and $f$ is zero on the boundary. 


Let $$ \Omega\_{\delta} = \{ x\in\Omega : \inf\_{y\in\partial\Omega} \left\|x-y\right\|\_2 > \delta \} .$$ Where $\delta>0$ is small enough to preserve smoothness in the boundary of $\Omega\_{\delta}$. See: [Shrinking a Lipschitz smooth domain.](https://mathoverflow.net/questions/46628/shrinking-a-lipschitz-smooth-domain).


For sufficiently small $\delta>0$ it is true that:


$\left\|f\right\|\_{L\_2(\Omega\setminus\Omega\_{\delta})} \leq C\delta^\alpha\left\|f\right\|\_{L\_2(\Omega)}$ with $\alpha\geq 1$ and $C$ is a constant not depending on $\delta$ or $f$. 


If this is not possible what is the largest $\alpha\in(0,1)$ so that the above inequality holds. 


Thanks in advance.


Note this is almost identical to my previous post [Bounding a smooth function near the boundary](https://mathoverflow.net/questions/46650/bounding-a-smooth-function-near-the-boundary) but presented in a clearer fashion. I would be grateful for any comments or help. 



Bounding near the boundary for a Sobolev function.
I don't believe your inequality holds. Your proposed inequality depends only on the $L\_2$ norm of $f$. Since the Sobolev space is dense in $L\_2$, if your inequality were true, it would also extend to any $L\_2$ function as well. I am also fairly sure it is easy to construct counterexamples to your inequality. You need to use the Sobolev norm on the right side of the inequality and not just the $L\_2$ norm.


