From classic literature, I know the following result.
Let $\Omega\subset\mathbb{R}^d$ be a bounded open set of class $C^1$. Then there exists $C>0$ such that \begin{equation}\label{1} \|\frac{u}{d} \|_{L^2(\Omega)}\le C\|\nabla u\|_{L^2(\Omega)}\qquad \forall\ u \in H^1_0(\Omega), \end{equation} where $d(x):=\operatorname{dist}(x,\partial\Omega)$.
I am wondering if a fractional order equivalent of the previous result holds, namely $$ \|\frac{u}{d} \|_{H^s(\Omega)}\le C\|\nabla u\|_{H^s(\Omega)}\qquad \forall\ u \in H^1_0(\Omega)\cap H^2(\Omega), $$ for every $0\le s<\frac{1}{2}$.
Does anyone have any ideas or counterexamples?
Thanks!